THE  CONSTRUCTION 

OF  , 

GRAPHICAL  CHARTS 


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THE  CONSTRUCTION  OF 

GRAPHICAL  CHARTS 


BY 

JOHN  B.  PEDDLE 

PROFESSOR  OF  MACHINE  DESIGN,   ROSE  POLYTECHNIC  INSTITUTE 


SECOND  EDITION 
REVISED  AND  ENLARGED 


McGRAW-HILL  BOOK  COMPANY,  INC. 

239  WEST  39TH  STREET.    NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  Lm 
6  &  8  BOUVERIE  ST.,  B.  C. 

1919 


ft 


RAW;!!!^  .BQOK,  ^OM^ANY,  .lye 


THJC    MA.PLB    PHE8S    YORK    PA 


PREFACE  TO  THE  SECOND  EDITION 


The  increased  use  of  charts  which  has  been  so  noticeable  within 
the  few  years  which  have  elapsed  since  this  book  was  published,  is  good 
evidence  of  the  growing  recognition  of  the  value  of  Nomography  to 
the  engineer. 

Improved  methods  for  constructing  charts  are  more  than  ever  desirable, 
and  I  therefore  welcome  the  opportunity,  which  comes  with  the  publi- 
cation of  a  new  edition  of  the  book,  to  add  a  chapter  describing  a  method 
which  has  been  very  successful  in  my  hands,  and  which,  I  trust,  will  be 
equally  so  in  those  of  my  readers. 

The  word  "Determinants"  in  the  heading  for  Chapter  VIII  is  the 
only  drawback  to  the  process,  but  I  hope  that  no  one  will  be  deterred  by 
the  name  from  giving  the  method  a  fair  trial.  I  have  used  it  for  a  number 
of  years  with  classes  in  this  subject,  and  my  experience  with  them  leads 
me  to  believe  that  anyone  with  the  ordinary  knowledge  of  mathematics 
which  an  engineer  should  possess  can  understand  and  apply  the  process 
with  but  little  difficulty. 

JOHN  B.  PEDDLE. 

January,  1919. 


493004 


PREFACE  TO  THE  FIRST  EDITION 


Much  of  the  work  of  calculation  done  by  the  engineer  or  designer  is  In 
the  repeated  application  of  a  limited  number  of  formulas  to  a  variety  of 
different  conditions,  which  involves  merely  the  substitution  of  different 
variables  in  identical  equations. 

Any  mechanical  means  for  performing  this  operation  expeditiously 
will  not  only  lead  to  a  saving  of  time  and  mental  wear  and  tear,  but  will 
also  minimize  the  chances  for  error. 

Such  a  device  is  the  calculating  chart,  or  nomogram,  and  the  increasing 
frequency  with  which  it  is  employed  in  the  more  recent  technical  publica- 
tions is  a  good  evidence  of  the  growing  recognition  of  its  value. 

Many  excellent  examples  of  these  charts  have  appeared  of  late  years 
and  are  available  for  use,  but  it  is  evident  that  to  realize  their  full  value 
as  useful  instruments  the  engineer  should  have  a  sufficient  acquaintance 
with  their  underlying  principles  to  construct  charts  suited  to  his  individual 
needs. 

Some  of  the  chart  forms  employed  to-day  have  been  known  and  used 
for  many  years,  but  it  is  only  within  recent  times  that  any  systematic 
study  has  been  made  of  the  subject  as  a  whole  or  any  attempt  to  properly 
classify  and  correlate  the  different  types. 

In  this  work  the  French  have  been  pioneers,  and  it  is  to  one  of  them, 
Maurice  d'Ocagne,  that  we  owe  what  is  probably  the  most  thorough  and 
comprehensive  text  on  the  subject,  his  "Traite  de  Nomographie." 

Although  books  on  nomography  have  been  published  in  many  foreign 
languages,  there  does  not  appear  to  have  been  anything  written  on  the 
subject  in  English  outside  of  a  few  scattered  magazine  articles  which 
have  covered  only  restricted  portions  of  the  field.  Books  in  English  on 
graphical  calculus  and  computation  are  by  no  means  uncommon,  but 
this  is  generally  looked  upon  as  something  different  from  nomography, 
although  a  strict  line  of  demarcation  between  the  two  subjects  would  be 
somewhat  difficult  to  trace. 

It  was  with  the  idea  of  supplying  an  elementary  English  text  in  this 
neglected  field  that  the  following  chapters  (originally  contributed  in 
serial  form  to  the  American  Machinist)  were  written, 

vii 


Vlii  PREFACE. 

Believing  that  the  subject  should  be  particularly  useful  to  the  practising 
engineer,  who  is  often  a  trifle  rusty  in  some  parts  of  his  mathematics,  an 
effort  has  been  made  to  simplify  the  mathematical  treatment.  A  series 
of  illustrative  problems  has  also  been  worked  out  in  detail  for  nearly  all 
the  chart  forms  which  are  here  described,  as  it  was  thought  that  a  study  of 
these  would  afford  a  clearer  insight  into  the  methods  and  a  better  under- 
standing of  the  difficulties  likely  to  be  encountered  than  would  be  possible 
from  a  purely  theoretical  analysis. 

The  desire  for  simplicity  in  mathematical  treatment  has  made  it 
necessary  to  restrict  the  application  of  the  charts  to  the  simpler  forms  of 
equation.  Equations  of  the  more  complex  types  may  be  and  have  been 
charted,  but  the  mathematical  difficulties  are  such  as  to  make  a  discussion 
of  the  methods  used  out  of  place  in  the  present  volume. 

The  processes  described  here,  if  thoroughly  understood,  should  be 
sufficient  to  cover  a  large  proportion  of  the  formulas  in  common  use. 
Those  of  my  readers  who  wish  to  pursue  the  subject  further  are  referred 
to  the  more  ambitious  works  of  d'Ocagne,  Soreau,  and  others. 

JOHN  B.  PEDDLE. 

August,  1910. 


CONTENTS. 


CHAPTER  I. — CHARTS  PLOTTED  ON  RECTANGULAR  CO-ORDINATES,       i 
The  simplest  form  of  chart.     Charts  plotted  on  rectangular 
coordinates.      Chart    for    the    proportions   of    band   brakes. 
Charts  with  irregular  scales.     Chart  for  focal  distance  of  a  lens. 
Logarithmic  charts. 

CHAPTER  II.— THE  ALINEMENT  CHART 15 

The  alinement  chart.  Chart  for  areas.  Chart  for  collapsing 
pressure  of  tubing.  Chart  for  twisting  moment  of  a  shaft. 
Doubled  or  folded  scales.  Alinement  chart  with  curved 
support. 

CHAPTER  III. — ALINEMENT  CHARTS  FOR  MORE  THAN  THREE 

VARIABLES 31 

Chart  for  helical  compression  spring.  Chart  for  strength  of 
gear  teeth.  Chart  for  strength  of  rectangular  beam. 

CHAPTER  IV.— THE  HEXAGONAL  INDEX  CHART  .    .    .    .    .    .    .     43 

The  hexagonal  index  chart.  Modification  of  the  preceding 
type. 

CHAPTER  V.— PROPORTIONAL  CHARTS 48 

The  proportional  chart.  Chart  for  strength  of  thick  hollow 
cylinders.  The  rotated  proportional  chart.  Chart  for  re- 
sistance of  earth  to  compression.  Charts  with  parallel  axes 
for  sums  or  differences.  Chart  for  centrifugal  force.  Chart 
for  piston-rod  diameter.  The  Z-chart.  Chart  for  polar  mo- 
ment of  inertia.  Chart  for  intensity  of  chimney  draft.  Chart 
for  safe  load  on  hollow  cast-iron  columns. 

ix 


X  CONTENTS. 

PAGE 

CHAPTER  VI.— EMPIRICAL  EQUATIONS 68 

Empirical  equations.  Finding  the  equation  of  a  straight 
line.  Another  illustration  of  finding  the  equation  of  a  straight 
line.  Finding  the  equation  of  a  curve.  Method  of  selected 
points.  Another  illustration  of  the  method  of  selected  points. 
Value  of  logarithmic  cross-section  paper  in  determining  form 
and  constants  of  an  equation.  Method  of  areas  and  moments. 
An  alinement  chart  method.  Another  illustration  of  the 
alinement  chart  method. 

CHAPTER  VII. — STEREOGRAPHIC  CHARTS  AND  SOLID  MODELS    .     98 
Three  dimensional  charts.     Axonometric  charts.     The  solid 
model.     Cardboard  substitute  for  solid  model.     The  tri-axial 
model. 

CHAPTER  VIII.—  THE  USE  OF  DETERMINANTS no 

Elementary  principles.  A  "  Z  "  or  "  N  "  type  chart  for  the  section 
modulus.  A  modification  of  the  previous  chart.  A  chart  for  the 
Francis  weir  formula.  A  hydraulic  chart  having  only  two  scales. 
A  logarithmic  chart  for  the  modulus  of  resilience.  A  four  variable 
chart  for  the  energy  of  a  moving  mass.  A  four  variable  chart 
for  Lame's  formula  for  thick  cylinders.  Four  or  more  variable 
logarithmic  charts.  A  four  variable  logarithmic  chart  for  helical 
compression  springs.  A  four  variable  "Z"  chart  for  helical 
compression  springs.  A  parallel  alinement  chart  for  the  Bazin 
formula.  Special  methods  for  refractory  equations.  Chart  for 
spherical  distance.  Chart  for  brake  bands.  Concluding  remarks. 


CONSTRUCTION  OF 

GRAPHICAL    CHARTS 


CHAPTER  I. 

CHARTS  PLOTTED  ON  RECTANGULAR  CO-ORDINATES. 
THE  SIMPLEST  FORM  OF  CHART. 

The  simplest  form  of  graphical  chart  is  that  which  is  frequently  used 
to  compare  different  systems  of  units  of  the  same  character  with  each 
other.  It  is  often  used,  for  instance,  to  show  the  relative  values  of  tem- 
peratures as  measured  on  the  Centigrade  and  Fahrenheit  scales. 

It  is  exceedingly  simple  to  construct  and  to  use. 

If  an  equation  containing  but  one  variable  and  its  function  is  to  be 
represented,  one  side  of  a  straight  line  is  graduated  to  represent  one  of  the 
variables,  and  the  equation  solved  to  give  as  many  corresponding  values 
of  the  other  variable  as  are  needed.  These  are  laid  off  on  the  other  side 
of  the  line,  and  in  order  to  read  the  chart  we  have  merely  to  run  across 
the  line  from  one  scale  to  the  other  to  get  corresponding  values  of  the 
variables.  It  may  be  used  for  a  variety  of  equations,  such  as  y  =  ax+b, 


v  =  \/2  g  h,  s  =  1  1  2  a  /2,  a  =  —  d2,  y  =  log.  x,  y  =  sin.  x,  etc. 

4 

For  purposes  of  illustration  I  have  plotted  the  two  charts  shown  in  Fig. 
i  to  represent  the  corresponding  values  of  the  diameter  and  area  of  the 
circle.  Such  a  chart  is  of  very  little  practical  value,  since  a  table  of  circular 
areas  will  give  the  desired  results  with  much  greater  accuracy  and  con- 
venience. I  have  introduced  it  here  partly  to  illustrate  the  type  of  chart, 
but  mainly  for  the  purpose  of  discussing  the  relative  merits  of  the  two 
systems  of  graduation  which  are  shown. 

It  will  be  noted  that  in  Chart  A  the  diameters,  are  expressed  in  equal 
scale  divisions,  and  the  areas  by  divisions  which  diminish  in  size  as  the 
areas  increase.  In  B  the  areas  are  represented  by  equal  divisions  and  the 
diameters  by  divisions  which  increase  in  size  as  the  diameters  increase. 

I     " 


OF   GRAPHICAL  CHARTS 


The  accuracy  with  which  we  can  read  such  charts  will  evidently  depend 
upon  the  size  of  the  divisions.  In  general,  the  conditions  represented  in  A 
are  preferable  for,  although  the  absolute  error  in  reading  the  upper  part 
of  the  unequal  scale  will  be  greater  than  in  the  lower  part,  the  percentage 
of  error  throughout  the  scale  will  be  more  nearly  equal  with  A  than  with  B. 

0  1  2  3  4  5    '  6    Diameter 

3zn±p=n±|i,i  I  li  1,11)1  I  i  J  l.i  i, i  i  l,i  1,1,1  1,1,11  1,1,1,1  I  rtrmd 


1  23456789  10  15  20  25  Area 

A 

15  2  2.5  Diameter 

— i — i — i — i — i — i — I — i 1 — i       i        I        i        i    H 


i  1 1  1 1 1  i  i  I  i  i  i  i  I  i  i  i  i  | 

0123456    Area 

B 

FIG.  i. — Plotted  scales  of  the  diameter  and  area  of  circles. 

On  the  other  hand,  if  most  of  our  readings  are  to  be  about  the  upper 
part  of  the  scale,  it  may  pay  us  to  use  the  B  arrangement  in  order  to  take 
advantage  of  its  larger  divisions. 

CHARTS  PLOTTED  ON  RECTANGULAR  CO-ORDINATES. 

Let  us  take  an  equation  of  the  form 

y  =  b±aoc.  (i) 

This  equation,  when  plotted  on  rectangular  coordinates,  gives  us  a 
straight  line.  That  is,  if  we  lay  off  values  of  y  on  the  vertical  or  Y-axis 
and  of  x  on  the  horizontal  or  X-axis  and  erect  perpendiculars  to  these  axes 
at  corresponding  values  of  x  and  y,  these  perpendiculars  will  intersect  at 
points  which  lie  on  the  same  straight  line.  Thus  in  Fig.  2,  line  7  corre- 
sponds to  the  equation 

;y=io+i/2  x. 

If  we  erect  a  perpendicular  to  any  point  on  the  X-axis,  say  40,  find  its 
intersection  with  line  7,  and  then  run  horizontally  to  the  Y-axis,  we  will  get 
the  corresponding  value  of  y  as  30. 

If  we  give  b  different  values,  say  15,  20,  and  25,  leaving  a  the  same,  we 
get  the  parallel  lines  6,  5,  and  4,  which  intersect  the  Y-axis  in  the  new 
values  of  b.  If  we  change  a,  we  change  the  slope  of  the  line;  if  we  make  it 
negative,  we  get  the  downward  sloping  lines  3,  2,  and  i. 

Suppose  we  make  a  in  the  equation  equal  to  i.     Our  sloping  lines  will 
now  run  at  an  angle  of  45  degrees.     Taking  a  new  chart  to  avoid  confusion' 
we  will  have  something  like  Fig.  3.     Two  sets  of  diagonals  are  shown:  one 


CHARTS    PLOTTED    ON    RECTANGULAR    CO-ORDINATES 


sloping  up  as  we  move  to  the  right  and  the  other  sloping  down.  The  first 
corresponds  to 

y  =  b+x  (2) 

and  the  other  to 

y  =  b-oc  (3) 

According  to  the  first  equation,  we  have  y  as  the  sum  of  b  and  x.  If, 
therefore,  we  enter  on  the  X-axis  at,  say  24,  run  up  as  indicated  by  the 
heavy  line  to  diagonal  15,  and  thence  to  the  Y-axis,  we  will  read  the  sum,  or 
39.  Subtraction  would  be  performed  by  going  in  the  opposite  direction  or 
by  using  the  b  lines  designated  by  negative  values. 


50 


40 


20 


10 


10 


20 


30 


40 


50 


FIG.  2. — Lines  plotted  from  the  general  equation  y=b±ax. 

Right  here  it  might  be  well  to  suggest  that  the  quantities  represented 
by  the  diagonal  lines  in  this  or  any  other  chart  should  be  such  as  are  not 
likely  to  vary  much,  and  are  capable  of  being  expressed  in  round  numbers. 
Fractional  values  can  be  much  more  easily  picked  off  of  the  scales  on  the 
axes.  A  large  number  of  diagonals  on  the  drawing  is  very  likely  to  cause 
confusion  in  reading,  and  will  certainly  entail  additional  labor  to  construct. 

Let  us  now  consider  the  other  set  of  diagonals,  corresponding  to 

y  =  b  —x. 
This  may  also  be  written 

b=x+y.  (4) 


4  CONSTRUCTION   OF   GRAPHICAL   CHARTS 

It  indicates  that  if  we  enter  the  X-  and  Y-axes  with  two  numbers  to  be 
added  and  run  the  perpendiculars  out  to  their  intersection,  this  intersec- 
tion will  be  found  on  the  diagonal  numbered  with  the  sum.  Thus  entering 
the  X-  and  Y-axes  at  26  and  44,  and  running  as  indicated  by  the  heavy 
lines,  we  find  the  intersection  on  diagonal  70.  Next  suppose  b  =  o,  and 
give  a  different  values.  The  diagonals  will  now  be  a  series  of  radiating 
lines  from  the  intersection  of  the  X-  and  Y-axes.  This  is  shown  in  Fig.  4. 


50 


4U 


20 


10 


0  10  20  30  40  50 

FIG.  3. — Lines  plotted  from  the  general  equation  y=b±.x. 

Here,  as  our  equation  informs  us,  the  chart  may  be  used  for  multiplication. 
Entering  the  X-axis  at  2,  running  up  to  the  diagonal  3,  and  from  there  to 
the  Y-axis,  we  read  the  product,  6.  Division  is,  of  course,  performed  by 
going  through  the  chart  in  the  opposite  direction. 

This  chart,  while  simple  in  appearance,  is  not  very  practical  where 
the  multipliers  differ  greatly  in  value.  It  is  easily  seen  that  if  we  wish  to 
multiply  any  number  on  the  X-axis  by  10,  it  will  be  necessary  to  have  the 
chart  10  times  as  high  as  it  is  wide.  Moreover,  the  intersection  of  the 
vertical  lines  with  the  diagonals  near  the  lo-line  is  very  acute  and  neces- 
sarily difficult  to  read  accurately.  The  best  position  for  the  diagonal  for 
purpose  is  on  or  near  the  45-degree  angle. 


CHARTS   PLOTTED    ON  RECTANGULAR   CO-ORDINATES  5 

These  difficulties  may  be  partly  overcome  by  changing  the  scale  values. 
If  we  renumber  the  diagonals  from  o .  i  to  i  making  their  values  10  times 
as  great,  as  shown  in  the  parentheses,  and  also  give  the  graduations  on 
the  Y-axis  a  double  set  of  numbers,  we  may  be  able  to  keep  the  dimensions 
of  the  chart  within  reasonable  limits  and  also  use  diagonals  which  are 
more  favorably  disposed  for  accurate  reading.  In  any  case,  however, 
there  will  be  an  unavoidable  crowding  together  of  the  diagonals  near  the 


123456789 
FIG.  4. — Lines  plotted  from  the  equation  y  =  ax. 


10 


origin  which  will  make  the  readings  about  the  low  numbers  difficult,  if 
not  impossible. 

There  was  no  real  need  to  suppose  that  b  in  the  equation  was  zero.  It 
was  done  merely  for  convenience  in  illustrating  the  point  I  wished  to 
explain.  If  b  had  had  any  value,  positive  or  negative,  we  should  have 
had  the  same  set  of  radiating  lines,  but  their  point  of  intersection  would 
have  been  shoved  up  or  down  the  Y-axis  by  the  value  which  we  give  to  b. 

Let  us  now  investigate  another  form  of  chart  for  multiplying,  writing 
our  equation 

a=xy  (5; 

If  we  give  a  a  definite  value  and  find  corresponding  values  of  x  and  y, 


6  CONSTRUCTION   OF  GRAPHICAL  CHARTS 

it  will  be  found  that  perpendiculars  erected  at  these  corresponding  points 
will  intersect  on  a  curve  called  the  equilateral  hyperbola.  For  each 
different  value  of  a  we  will  have  a  different  curve. 

A  chart  constructed  with  them,  like  Fig.  5,  could  therefore  be  used 
for  multiplication  and,  of  course,  for  its  converse,  division.  We  have 
only  to  pick  out  the  numbers  to  be  multiplied  on  the  two  axes,  follow  up 
their  perpendiculars  to  their  point  of  intersection,  which  will  be  found 
on  the  curve  numbered  with  the  product.  Should  this  point  fall  between 
two  curves,  instead  of  on  one  of  them,  the  product  must  be  interpolated 


0  1  2        ~3  4  5  6-7  8  9          10 

FIG.  5.— Chart  for  multiplication  and  division,  plotted  from  the  equation  a=xy. 

by  eye.  It  will  readily  be  seen  that  this  method  is  not  at  all  suited 
to  any  case  in  which  the  desired  number  of  products  is  large,  since  the 
labor  of  drawing  in  the  curves  would  be  prohibitory. 

Note  that  curves  i  or  10  might  be  used  as  tables  of  reciprocals. 

Next,  let  us  consider  a  case  in  which  some  power  of  one  of  the  quanti- 
ties is  involved.  We  will  select  a  case  involving  several  multiplications 
in  order  to  show  how  some  of  the  principles  already  discussed  are  applied. 
This  will  be  done  in  some  detail  in  order  to  clearly  show  the  process  of 
attacking  a  simple  problem. 


CHARTS  PLOTTED  ON  RECTANGULAR  CO-ORDINATES 

CHART  FOR  PROPORTIONS  OF  BAND  BRAKES. 
Take  the  formula  for  the  band  brake 


in  which  P  represents  the  resultant  tangential  pull  on  the  brake,  A  the 
area  of  the  cross  section  of  the  brake  band  in  square  inches,  T  the  tension 
in  the  tight  side  of  the  band  in  pounds  per  square  inch,  /the  coefficient  of 
friction  (0.18  in  the  case  of  iron  on  iron)  and  a  the  arc  of  contact  of  the 
band  in  degrees. 

Inside  of  the  parenthesis  in  our  equation  there  is  only  one  quantity 
which  need  be  considered  as  a  variable,  a  the  arc  of  contact;  /  will  be 
constant  for  any  given  materials  for  band  and  drum  and,  as  indicated 
above,  will  be  taken  as  0.18.  Under  these  circumstances,  instead  of 
drawing  a  separate  line  or  set  of  lines  for  each  quantity  inside  the  parenthe- 
sis, we  need  only  draw  one  line  for  the  parenthesis  as  a  whole,  getting  the 
different  values  for  plotting  this  line  by  letting  a  vary.  We  will  have  to 
assume  the  limits  within  which  this  variation  is  to  take  place.  Suppose 
we  take  these  as  200  and  300  degrees.  Then  solve  the  parenthesis  for 
every  10  degrees  between  these  limits. 

In  Fig.  6  the  results  of  these  calculations  are  shown  plotted  as  ordinates 
on  the  chart,  while  the  corresponding  arcs  of  contact  are  taken  as  abscissas. 
In  laying  off  the  latter,  one  small  scale  division  on  the  horizontal  scale  is 
used  to  represent  two  degrees  of  arc.  The  vertical  scale  will  need  to  be 
large  as  the  values  of  the  parenthesis  only  vary  from  0.4666  to  0.6103, 
and  this,  if  plotted  to  a  small  scale,  would  make  a  very  flat  and  therefore 
undesirable  curve. 

Suppose  we  make  one  small  scale  division  on  the  vertical  scale  equal  to 
o.  01.  This  has  been  done  on  the  chart,  and  the  curve  drawn  through  the 
points  thus  found.  These  values  must  now  be  multiplied  by  the  assumed 
values  of  71,  the  tension  per  square  inch  in  the  band.  According  to  one 
authority,  the  safe  values  for  T  will  range  from  4500  to  6500  for  wrought 
iron,  and  from  8500  to  11,500  for  steel.  We  have  therefore  to  provide  for 
a  total  range  of  7000  pounds  and  we  will  cover  this  by  steps  of  500  pounds. 
We  will  adopt  the  multiplying  method  shown  in  Fig.  4,  making  the  radiat- 
ing lines  stand  for  the  different  tensions.  They  must  converge  to  a  point 
somewhere  on  the  zero  line  of  the  curve  just  drawn,  and  this  point  may 
be  chosen  at  will.  In  reading  the  chart  we  must  run  up  or  down  a  vertical 
line  until  we  strike  the  curve,  and  then  go  horizontally  until  we  reach  the 
desired  T-line.  It  is  evident  that  all  the  T'-lines  must  be  in  such  a  position 
that  they  may  be  intersected  by  any  horizontal  drawn  from  the  curve. 


8 


CONSTRUCTION  OF  GRAPHICAL  CHARTS 


They  must  be  so  drawn  that  the  tangents  of  the  angles  they  make  with  the 
vertical  will  be  proportional  to  the  tensions  they  represent.  Let  us  run 
up  ten  of  the  large  divisions  from  the  zero  line  and  then  horizontally  41/2, 
5,51  /2,  6,  etc.,  of  the  large  divisions,  corresponding  to  tensile  stresses  of 

Arc  of  Contact  in  Degrees 
200°     210°       220°      230°       240°      250°      260°       270°    "  280°       290'      300" 


200 


0   3400 


FIG.  6. — Proportions  of  band  brakes. 


4500,  5000,  5500,  6000,  etc.,  so  as  to  get  the  lines  well  spread  out.  If  we 
take  the  point  of  convergence  at  14  large  divisions  to  the  right  of  the  left- 
hand  edge  of  the  chart,  the  conditions  we  have  imposed  above  will  be 
fulfilled,  and  this  has  accordingly  been  done.  The  results  of  this  multipli- 


CHARTS    PLOTTED    ON   RECTANGULAR    CO-ORDINATES  9 

cation  will  be  read  on  some  horizontal  axis,  and  they  must  next  be  multi- 
plied by  the  assumed  values  of  Ay  the  area  of  the  cross  section  of  the  band. 

We  could  use  the  same  point  of  convergence  for  the  A  -lines  as  for  the 
T-,  but  inasmuch  as  this  would  cause  some  confusion  in  reading  the  dia- 
gram, it  will  be  better  to  use  some  other  center,  which,  however,  must  be 
located  on  the  vertical  line  passing  through  the  T  center.  According  to 
the  authority  quoted  above,  the  thickness  of  the  band  for  ordinary  cases 
should  vary  between  0.08  inch  and  o.  16  inch,  corresponding,  roughly,  to 
No.  12  and  No.  6  Brown  &  Sharpe  gage.  If  our  bands  are  not  to  be 
less  than  i  inch  nor  more  than  3  inches  in  width,  the  maximum  variation 
in  area  will  be  between  0.08  square  inch  and  0.48  square  inch.  For 
convenience  let  the  areas  vary  by  steps  of  o .  04  square  inch,  although  any 
other  size  of  step  might  have  been  chosen.  This  will  give  us  n  lines 
which  must  be  so  drawn  that  the  tangents  of  the  angles  they  make  with 
the  horizontal  will  be  proportional  to  0.08,  o.  12,  o. 1 6,  etc. 

We  must  now  determine  the  limits  within  which  our  results,  the  desired 
values  of  P,  must  fall.  For  the  least  area,  0.08  square  inch;  the  least 
tension,  4500;  and  the  smallest  contact  angle,  200  degrees,  we  have  P=i68. 
For  the  largest  values  of  the  same  quantities  we  have  P  =  ^6g.  These 
values  of  P  will  be  read  on  a  vertical  scale.  It  will  be  found  that  if  we 
allow  i  large  division  on  the  vertical  scale  to  represent  200  pounds  it 
will  give  us  a  convenient  scale  length  and  readings  may  be  made  with  an 
accuracy  which  is  sufficient  for  all  practical  purposes.  The  length  of 
the  vertical  scale  will  thus  be  about  17  of  the  large  divisions. 

Therefore,  going  up  1 7  large  divisions  from  the  zero  of  the  curve,  we 
locate  the  center  for  the  radiating  A  -lines  on  the  vertical  line  which  passes 
through  the  center  of  the  JMines.  From  this  center  we  go  10  large  divi- 
sions to  the  left  and,  going  down  2,  3,  4,  5,  etc.,  large  divisions  (propor- 
tional to  0.08,  o.  12,  o.i 6,  o. 20,  etc).,  we  locate  the  points  through  which 
the  A  -lines  must  be  drawn  from  their  center.  By  this  arrangement  they 
will  cover  the  desired  length  on  the  scale  of  P.  Our  chart  is  now  complete 
except  for  lettering  the  lines  and  scales.  The  left-hand  scale  must  of 
course  be  lettered  so  as  to  make  each  large  division  represent  200  pounds' 
pull. 

To  read  the  chart,  enter  at  the  bottom  or  top  at  the  assumed  arc  of- 
contact  and  run  up  or  down  to  the  curve,  from  there  go  horizontally  to  the 
desired  tension  in  the  band,  then  vertically  to  the  area  line,  and  then 
horizontally  to  the  vertical  scale  representing  the  tangential  pull.  Or,  if 
the  pull,  arc  of  contact  and  tension  are  known,  enter  as  before  at  the  arc 
of  contact,  run  vertically  to  the  curve,  thence  to  the  tension  line,  and  the 


10  CONSTRUCTION   OF   GRAPHICAL   CHARTS 

intersection  of  the  vertical  through  this  point  with  the  horizontal  drawn 
from  the  desired  pull  will  be  on  or  near  one  of  the  area  lines,  thus  giving 
the  necessary  size  of  the  band. 

It  is  obvious  that  for  all  practical  purposes  our  chart  might  have  been 
trimmed  off  on  the  right-hand  side  at  the  end  of  the  curve  so  as  to  omit 
all  of  the  diagram  not  sectioned  with  the  small  divisions,  also  that  there 
is  no  need  of  continuing  the  T-lines  above  or  below  the  curve. 

CHARTS  WITH  IRREGULAR  SCALES. 

There  is  no  necessity  in  these  charts  for  having  the  scale  divisions 
equal,  as  has  been  the  case  in  all  the  charts  except  the  first.  If  we  admit 
this,  there  is  a  distinct  advantage  in  many  cases  in  having  them  irregular. 

CHART  FOR  THE  FOCAL  DISTANCE  OF  A  LENS. 

For  instance,  take  the  formula  connecting  the  two  foci  of  a  lens  with  its 
principal  focus 


where  /  and  /'  are  conjugate  focal  distances  and  p  the  principal  focal 
distance. 
Make 

i  i          ,  ,       i 

*=7'y=r      y 

The  above  equation  becomes  x+y  =  b  which  is  identical  with  equation  (4) 
above. 

We  have,  in  this  case,  to  lay  out  on  the  X-  and  Y-axes  the  reciprocals 
of/  and/'  and  draw  in  the  diagonals  as  shown  in  Fig.  7,  just  as  we  did 
in  Fig.  3.  Knowing  the  principal  focal  distance  of  our  lens,  we  select  the 
diagonal  corresponding  to  it,  enter  the  X-axis,  say,  at  the  distance  of  the 
object  from  the  lens,  run  up  to  the  diagonal,  from  there  to  the  Y-axis,  and 
read  off  the  distance  at  which  the  object  will  be  in  focus. 

LOGARITHMIC  CHARTS. 

A  more  important  case  is  where  the  divisions  are  laid  off  to  a  logarith- 
mic scale.  Paper  ready  ruled  in  this  way  may  now  be  had  from  dealers 
in  mathematical  instruments  and  is  valuable  for  many  purposes.  On  it 
many  problems  which  would  have  to  be  solved  by  tediously  drawn  curves, 
may  be  worked  with  ease  by  straight  lines. 


CHARTS    PLOTTED   ON    RECTANGULAR    CO-ORDINATES 


II 


Let  us  return  to  equation  (5)  a=xy.  This  may  also  be  written  log.  a  = 
log.  x+log.  y  which  is  identical  with  (4),  the  equation  for  a  straight  line. 
The  paper  in  question  is  graduated  on  its  horizontal  and  vertical  axes 
so  that  the  lengths  from  the  origin  are  equal  to  the  logarithms  of  the  num- 
bers placed  opposite  the  graduation  marks. 

If  in  Fig.  8  we  connect  2  on  the  vertical  axis  with  2  on  the  horizontal 
axis,  3  with  3,  and  so  on,  we  get  a  chart  similar  to  Fig.  3,  which  was 
used  for  addition,  but  in  this  case  is  for  multiplication.  It  also  bears  some 
resemblance  to  Fig.  5,  the  equilateral  hyperbolas  used  there  being 
replaced  by  straight  lines.  To  use  the  chart  enter  at  the  X-  and  Y-axes 
with  the  numbers  to  be  multi- 
plied and  follow  out  the  perpen- 
diculars at  these  points  to  their 
point  of  intersection,  which  will 
be  found  at  the  diagonal  num- 
bered with  the  product. 

We  might  also  draw  the  diag- 
onals so  as  to  slope  upward  from 
left  to  right  instead  of  downward, 
as  shown  on  the  same  chart. 
This  is  identical  with  the  second 
form  of  addition  chart  of  Fig.  3, 
and  may  also  be  used  for  multi- 
plication. Thus,  entering  on 


tanre  at  which  Object  will  be  in  Focus 
3  §  5  |S  ?£•*£*£  ~  g  -01  g  ? 

\ 

\ 

\ 

V' 

k  •%>. 

* 

b 

.  >< 

\ 

S 

\ 

\\ 

\ 

\ 

\ 

\\ 

\ 

s. 

\\ 

\ 

\ 

\ 

V    \.   N 

s^^     N 

\  */ 

x\\ 

\ 

\ 

\ 

\ 

~x 

N 

s  \ 

\ 

| 

\ 
\ 

S 

\ 

\v 

s 

\\ 

Xs 

^\ 

V 

V 

N 

\N 

S 

§ 

s 

ss 

\ 

X 

\ 

'^ 

N^s 

\v 

Xx 

\ 

.  \    rv 

^v^ 

X..^ 

s 

\\ 

\\^ 

5    v   x 

ga 

1 

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, 

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^ 

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\s 

s  \ 

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IA^MV 

Distance  of  Object  from  Lens 


FIG.  7. — Chart  for  focal  distances  of  a  lens. 


the  X-axis  with  the  multiplicand  we  run  up  till  we  strike  the  diagonal 
numbered  with  the  multiplier,  and  thence  over  to  the  product  on  the 
Y-axis.  Such  paper  is  also  very  convenient  for  handling  equations  con- 
taining powers  and  roots  of  the  variables,  and  especially  where  these 
powers  and  roots  are  fractional. 

For  instance,  y—x2  may  be  written  log.  y  =  2  log.  x. 

This  indicates  that  a  line  drawn  so  that  its  tangent  with  the  horizontal 
is  2  could  be  used  for  squaring  numbers  on  the  X-axis,  or  conversely  for 
extracting  tjie  square  roots  of  numbers  on  the  Y-axis.  This  is  shown  in 
Fig.  9.  The  top  of  the  diagram  is  bisected,  and  a  line  drawn  to  this 
point  from  the  origin,  enabling  us  to  find  any  square  not  exceeding  10. 
Entering  at  2  on  the  X-axis  and  running  up  till  we  strike  this  line  and 
from  there  to  the  Y-axis,  we  read  22,  or  4. 

To  get  squares  greater  than  10  we  should  have  to  extend  our  chart 
above  the  lo-line.  It  would  be  exactly  similar,  however,  to  the  part 
below,  and  it  is  therefore  only  necessary  to  lower  our  squaring  line  so  as  to 


CONSTRUCTION   OF  GRAPHICAL  CHARTS 
2  3  4567 


H 


2  3  4  56789    10 

FIG.  8. — Logarithmic  chart  for  multiplication. 


1  2  3  4,  5          6        7       8      9    10 

FIG.  9. — Lines  of  powers,  roots,  etc.,  on  logarithmic  paper. 


CHARTS    PLOTTED    ON    RECTANGULAR   CO-ORDINATES  13 

cut  the  base  of  the  chart  in  the  middle,  and  make  it  pass  through  the  upper 
right-hand  corner.  We  thus  get  a  chart  which  may  be  used  for  getting 
the  square  or  square  root  of  any  number,  the  only  thing  to  be  noted  in  the 
latter  operation  is  that  we  must  use  one  or  the  other  section  of  the  line 
according  to  the  position  of  the  decimal  point.  If  the  number  whose 
square  root  is  desired  has  one,  or  three,  or  five  places  (any  odd  number) 


Values  of  h 


5  6          7        8       9     10 


bk* 


Fig.  10. — Logarithmic  charts  plotted  from  the  equation  Z==— 

before  the  decimal  point,  use  the  first  section  of  the  line;  if  it  has  two,  or 
four,  or  six  places  (any  even  number),  before  the  decimal  point,  use  the 
second  section. 

From  what  has  been  said  it  is  plain  that  the  cube  line  should  be  drawn 
by  dividing  the  upper  and  lower  edges  of  the  diagram  into  three  parts  so 
as  to  make  the  tangent  of  the  angle  of  slope  3.  Here  there  will  be  three 


14  CONSTRUCTION   OP  GRAPHICAL   CHARTS 

^  / 

lines  crossing  the  diagram.  For  getting  cube  roots  the  first  section  should 
be  used  where  the  number  of  places  before  the  decimal  point  is  i,  4,  or  7, 
etc.,  the  second  section  where  the  number  of  places  is  2,  5,  or  8,  etc., 
while  the  third  section  is  used  where  the  number  of  places  is  3,  6,  or  9,  etc. 

For  getting  fractional  powers  or  roots  the  tangent  of  the  angle  of 
the  slope  must,  of  course,  be  equal  to  this  fractional  exponent.  Equa- 
tions such  as  pvn=c  are  easily  solved.  In  Fig.  9  the  line  representing 
pv*'**=  10  has  been  drawn  for  purposes  of  illustration,  v  being  read  on  the 
horizontal,  and  p  on  the  vertical  axis.  On  the  same  chart  a  line  has  been 
drawn  for  getting  circular  areas,  showing  the  extreme  simplicity  of  the 
method.  Diameters  are  read  on  the  horizontal  and  areas  on  the  vertical 
axis. 

In  Fig.  10  is  shown  the  application  of  this  paper  to  the  formula  for 
the  section  modulus  of  a  beam  of  rectangular  section 

bh> 
~~" 


In  this  chart,  values  of  h  are  read  on  the  base  line,  b  on  .the  diagonals, 
and  Z  on  the  vertical  or  Y-axis.  For  the  sake  of  clearness  only  two  of  the 
diagonals  representing  b  have  been  drawn.  They  are  for  b  =  2  and  b  =  4. 
The  intersections  with  the  vertical  or  Z-axis  are  found  by  letting  h=i. 
The  tangent  of  the  angle  of  slope  is  2. 

In  reading  any  of  the  logarithmic  charts  here  given,  significant  figures 
only  will  be  found.  No  definite  rules  need  be  given  for  finding  the  position 
of  the  decimal  point.  As  with  the  slide  rule,  it  needs  only  the  application 
of  a  little  common  sense. 


CHAPTER  II. 


THE  ALINEMENT  CHART. 

^  A  type  of  chart  which  has  received  considerable  attention  of  late  years 
and  which  differs  radically  from  those  already  described  is  that  known  as 
the  alinement  chart.  In  the  charts  hitherto  examined  the  necessary 
lines  were  plotted  on  what  are  known  as  rectangular  coordinates;  that 
is,  the  axes  on  which  the  values  of  x  and  y  were  plotted  met  at  a  right 
angle.  This  is  by  no  means  a  necessary  condition.  The  axes  may  be 
parallel,  and,  in  fact,  I  have  a  little  book  in  which  the  author  has  de- 
veloped a  system  of  coordinate  geometry  based  on  parallel,  instead  of 
rectangular  coordinates. 

To  aid,  us  in  understanding  this  form  of  chart,  let  us  take  an  equation 
of  the  form 

au+bv  =  c  (6) 

where  u  and  v  are  variables  and  a,  b,  and  c  are  constants.     This  is  the 
equation  of  a  straight  line  where  rectangular  coordinates  are  used.     To 
illustrate,  let  us  assume  a  =  4,  b  =  6, 
and  c  =  60,  and  draw  the  line  repre- 
sented by  the  equation  as  shown 
in  Fig.  ii. 

Since  u  and  v  may  have  any 
values,  let  u=o,  then  6  ^=60,  and 
v=io.  Again,  letting  v—o,  4u  = 
60,  and  ^=15.  This  gives  us  the 
coordinates  of  two  points  on  the 
line,  one  on  each  axis.  Lay  off-w=i5  on  the  Y-axis  and  v=io  on  the 
X-axis  and  join  them.  Then  for  v=4,  u=g,  as  shown  by  the  heavy  line 
on  the  chart. 

Now  let  us  lay  off  the  same  quantities  on  the  parallel  axes  in  the 
second  chart  of  Fig.  n.  On  the  axis  marked  A  u  lay  off  15  and  join  it 
to  v  =  o  on  the  B  v  axis.  Lay  off  v=io  on  the  B  v  axis  and  join  it  to 
u=o  on  the  A  u  axis.  These  two  lines  meet  at  the  point  marked  p.  It 
will  be  found  that  all  lines  joining  corresponding  values  of  u  and  v,  as 
found  from  the  equation,  will  pass  through  the  same  point.  Or,  if  we 


FIG.  n. — Comparison  of  a  rectangular 
and  alineme.nt  chart. 


l6  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

take  4  on  the  B  v  axis  and  join  it  with  />,  this  line  prolonged  will  cut  A  u 
Sit  9,  giving  us  the  same  result  that  we  got  by  the  other  chart. 

Thus,  what  was  the  equation  of  a  line  with  the  rectangular  system 
becomes  now  the  equation  of  a  point.  Keeping  a  and  b  constant  and 
changing  c  gives  us,  with  the  rectangular  system,  a  series  of  parallel 
lines-.  With  the  parallel  coordinates  this  merely  moves  the  point  p  up  or 
down  on  the  line  C  p."  This  line  is  called  the  "support"  for  the  points 
of  intersection  p.  On  the  other  hand,  changes  of  a  or  b  will  shift  p  to  the 
right  or  left  of  the  support  C  p.  To  establish  the  point  p  it  will  generally 
be  sufficient  to  solve  the  equation  for  a  few  easily  determined  values  of  u 
and  v,  lay  them  off  on  their  axes,  and  join  corresponding  points,  as  has 
just  been  done.  Or  it  may  be  worked  out  analytically  as  follows: 

Let  the  position  of  the  point  be  supposed  to  be  located  by  reference 
to  rectangular  coordinates,  of  which  the  line  A  B  represents  the  X-axis, 
and  the  line  through  O  midway  between  A  u  and  B  v  and  parallel  to  them 
the  Y-axis.  Draw  a  horizontal  line  through  p.  It  will  intersect  the 
lines  A  u  and  B  v  at  the  same  height  above  A  B  as  p.  Call  this  distance  y. 

Equation  (6)  may  now  be  written 

au+b  v  =  a  y+b  y  =  Cj 
or 


This  gives  the  distance  of  the  point  p  above  A  B. 

If  x',  y',  and  x"  y"  are  the  rectangular  coordinates  of  two  known  points 
on  a  line,  analytic  geometry  teaches  us  that 

x-x'  =  y-y 

x'—x"    y'  —  y"' 

Let  us  apply  this  to  the  two  lines  originally  drawn  to  locate  p,  and  call  the 
distances  O  A  and  O  B,—  d  and  +  d,  respectively.  The  coordinates 
of  the  points  at  the  two  ends  of  one  of  tfre  lines  are : 

c 

x'=  —  d,  y'  =  — ,   and  x"  =  d,  y"  =  o. 
a 

Therefore 

c 

x  4-  d  a 


—  d—d       c 

o 

a 

For  the  other  line  x'  =  —  d,  y'  =  o,  and  x"  =  d,  /'  =  -  -  are  the  co- 

o 


THE  ALINEMENT   CHART  1  7 


ordinates  of  the  ends,  and  its  equation  will  be 

x  +  d          y—o 
—  d—d  c 

°-y 

Combining  these  equations  so  as  to  eliminate  y  we  get 


thus  giving  us  the  distance  of  the  point  from  the  vertical  line  through  O. 
It  shows,  also,  that  this  distance  is  independent  of  c,  and  that,  therefore, 
however  c  varies  (if  a  and  b  are  constant)  ,  p  will  always  lie  on  the  line  C  p. 

The  actual  location  of  the  various  points  on  this  line  may  be  found 
by  solving  the  equation  for  y  for  different  values  for  a  and  6,  or,  as  said 
before,  by  joining  up  corresponding  points  on  the  A  u  and  B  v  axes  by 
lines  whose  intersection  with  the  locus,  or  support,  of  p  will  give  us  the 
desired  points. 

A  practical  example  worked  through  will,  perhaps,  give  us  a  better 
idea  of  the  methods  used  than  we  should  gain  by  a  purely  abstract 
discussion. 

CHART  FOR  AREAS. 

•  A  man  engaged  in  making  blueprints  asked  me  to  make  him  a  chart 
for  calculating  the  areas  of  his  prints  in  order  to  aid  him  in  fixing  his 
charges.  As  it  makes  a  very  simple  problem  for  this  purpose  I  will 
use  it  in  this  demonstration,  merely  remarking  that  the  diagram  furnished 
him  differs  in  some  particulars  from  the  one  used  for  illustration  here, 
in  order  to  better  adapt  it  to  his  needs. 
The  formula  used  was 


144 

in  which  A  is  the  area  in  square  feet,  W  the  width  in  inches  and  L  the 
length  in  inches. 

Write  this  in  its  logarithmic  form 

log.  W  +  log.  L  =  log.  A  +  log.  144. 

Let  us  plot  log.  W  on  the  A  u  axis,  log.  L  on  the  B  v  axis  and  log.  A 
on  the  intermediate  support. 

We  must  first  decide  upon  the  scales  by  which  these  lengths  are  to  be 
measured  on  their  axes.     For  instance,  u,  the  measured  length  of  log.  W 
on  the  A  u  axis,  is  obtained  by  multiplying  log.  W  by  some  "modulus" 
a 


i8 


CONSTRUCTION   OF   GRAPHICAL   CHARTS 


or  coefficient  to  get  the  desired  length  in  inches.     Let  us  call  this  modulus 
/j  for  the  A  u  axis,  12  for  the  B  v  axis,  and  /3  for  the  intermediate  support. 
Then 

u  =  I,  log.  W, 
v  =  12  log.  L, 
and 

log.  W  =  f ,    ' 

l\ 

log.  L  =  ^-. 


6"- 
5"- 

4"- 


2- 


15 


5- 


.1- 


FIG.  12. — Alinement  chart  for  areas. 


Calling  log.  A  +  log.  144  =  c,  we  have 


u       v 

+=C' 


or 


12  u  +  /j  v  =  /t  /2  c. 
From  equation  (8)  we  have 

b-a  - 


p-5ft. 
4ft. 

i-3ft. 
^-2ft. 


"8 


10' 


THE  ALINEMENT   CHART  19 

from  which 


12  ~  d-x        CB'  (9) 

thus  locating  the  support  for  the  product. 
From  the  equation  (7)  we  see  that 


= 


I*  +  V 
c  must  therefore  be  multiplied  by 


in  order  to  give  the  measured  lengths  along  the  third  axis,  or  support, 
and  this  quantity  must  be  its  modulus,  or 


h   +  (10) 

The  graduated  lengths  along  the  different  axes  may  be  anything  we 
choose  to  make  them.  In  general,  they  should  be  about  equal  and  as 
long  as  possible  while  keeping  the  size  of  the  chart  within  reasonable 
limits.  The  largest  size  of  print  which  was  called  for  in  this  problem 
was  40  x  60  inches.  The  logarithm  of  40  is  1.602,  and  of  60  is  1.778. 
These  numbers  are  so  nearly  equal  that  it  will  not  pay  us  to  use  different 
scales  in  laying  them  out  in  order  to  represent  them  by  exactly  equal 
lengths  on  the  axes,  and  lt  will  accordingly  be  made  equal  to  lv 

The  best  scale  to  use  for  a  practical  problem  would  probably  be  i 
inch  for  a  logarithmic  value  of  o  .  i  ,  thus  making  the  length  of  the  longest 
axis  about  18  inches.  In  the  drawing  made  for  this  article  the  twentieth 
scale  was  used,  giving  a  chart  half  this  size.  This  does  not  refer  to  the 
cut  which,  of  course,  has  been  reduced. 

Since  /t  and  12  are  to  be  equal,  equation  (9)  shows  that  the  distances 
C  A  and  C  B  are  equal,  or  the  support  for  the  areas  must  be  midway 
between  the  outside  axes.  For  the  modulus  on  the  third  axis  we  have 
from  equation  (10) 

/     *.*.    .'*. 

3  "  *,  +  *,  ==  « 

or  we  must  use  a  scale  of  fortieths  on  the  middle  axis,  on  which  the  areas 
are  plotted,  if  we  use  twentieths  on  the  outside  axes.  The  distance 
between  the  outside  axes  may  be  anything  we  wish.  If  the  axes  are 
too  close  we  get  a  compact  chart,  but  the  intersection  of  the  index  line 
with  the  axes  may,  in  some  positions,  be  so  acute  as  to  make  accurate 
reading  difficult.  The  farther  the  axes  are  apart  the  better  this  condition 


20  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

will  be,  but  we  must  not  make  the  distance  so  great  as  to  get  a  chart 
which  will  be  awkward  to  handle. 

Perhaps  the  best  arrangement  for  average  conditions  will  be  to  have 
the  chart  about  square,  in  which  case  the  index  line  will  never  make  a 
smaller  angle  with  the  axes  than  45  degrees;  this  is  not  objectionable. 

The  two  outside  axes  are  now  to  be  graduated  so  as  to  represent  the 
logarithms  of  the  desired  lengths  and  widths  expressed  in  inches.  Start 
with  i  inch  (whose  logarithm  is  o)  on  the  A  B  line. 

On  the  middle  axis  instead  of  putting  i  on  the  A  B  line  we  must 
remember  that  logarithm  A  is  to  be  added  to  logarithm  144  (which  is 
2.158),  and  we  therefore  run  up  21.58  measured  with  the  fortieth  scale 
before  beginning  to  graduate.  Calling  this  point  i,  we  lay  off  from  it 
the  logarithms  of  2,  3,  4,  etc.,  and  such  subdivisions  of  them  as  may  be 
necessary,  till  we  reach  17,  a  trifle  beyond  the  limits  of  our  other  scales. 
The  chart  is  now  complete  with  the  exception  of  the  lettering. 

To  read  it,  lay  a  straight-edge  or  draw  a  fine  thread  tightly  across  the 
chart  so  as  to  join  the  points  representing  the  length  and  width  of  the 
print,  and  the  intersection  of  the  line  with  the  middle  axis  will  give  the 
area  in  square  feet.  Better,  perhaps,  than  either  the  straight-edge  or 
thread  is  a  piece  of  glass  or  thin  celluloid  with  a  straight  line  scratched 
on  its  under  surface. 

Such  charts  as  this  will  ordinarily  show  a  very  marked  advantage 
over  those  previously  described.  They  are  usually  much  simpler  to 
construct,  and  they  avoid  the  confusing  tangle  of  lines  so  often  found 
with  the  rectangular  type.  Moreover,  since  we  do  not  have  to  draw  a 
separate  line  for  each  value  of  the  variable,  as  is  sometimes  necessary  with 
the  other  form,  it  will  be  easier  to  get  close  readings  by  interpolation. 

The  scope  of  this  chart  might  have  been  somewhat  enlarged,  without 
much  trouble,  had  it  been  thought  desirable.  The  prices  corresponding 
to  the  different  areas  might  have  been  marked  on  the  other  side  of  the 
area  line  in  something  the  same  manner  as  was  done  in  Fig.  i.  Also 
the  chart  might  have  been  extended  to  give  the  total  area  or  price  of  a 
number  of  prints  of  given  size.  To  do  this  we  should  merely  have  to 
consider  the  area  line  as  the  outside  axis  of  a  new  diagram,  the  other  out- 
side axis  being  graduated  to  represent  any  desired  number  of  prints,  and 
the  product  would  be  read  off  on  a  new  intermediate  axis.  The  A  B, 
or  base,  line  need  not  have  been  left  on  the  chart,  as  it  is  of  no  use  after 
the  construction  is  once  made,  and  it  will  generally  be  omitted. 


THE  ALINEMENT  CHART  21 

CHART  FOR  COLLAPSING  PRESSURE  OF  TUBING. 
Another  formula  charted  on  this  plan  is  shown  in  Fig.  13.     It  is 

/ 1  V 
P  =  50,210,000  (  — ), 

and  will  be  recognized  as  Stewart's  formula  for  the  collapsing  pressure 
for  bessemer-steel  tubing,  to  be  applied  to  pressures  not  exceeding  581 

pounds  or  to  values  of  —  not  exceeding  0.023. 

In  it  P  is  the  external  pressure  in  pounds  per  square  inch,  t  the  thick- 
ness of  the  tube  in  inches,  and  D  the  external  diameter  of  the  tube,  also  in 


600- 

^.13" 

• 

-.12" 

r 

500- 

-.11" 

r5" 

-.10" 

-.09" 

: 

400- 

"'h 

-.08" 

L4" 

I 

-.07" 
-.06"                     ^ 

: 

300- 

V. 

£-i                                                                                                                    C 

*                    1 

:3" 

3                                                                 J 

-.04"                    Jg 

r 

*''••"" 

IT    (/?»•                                                                 ^ 

I 

200- 

-.03" 

12" 

inn 

-.02" 

l" 

FIG.  13. — Alinement  chart  for  Stewart's  formula  for  collapsing  pressures 
of  Bessemer  tubing. 

inches.     It  is  very  similar  to  the  case  we  have  just  worked  out,  but  there 
are  one  or  two  practical  points  in  which  they  differ  which  will  make  it 
worth  our  while  to  hastily  run  through  the  construction. 
The  formula  may  also  be  written 

P  Dz  =  5o,2io,ooo/3 
or 

log.  P+3  log.  D  =  log.  50,210,000  +  3  log.  t. 

Its  essential  similarity  with  our  fundamental  equation  will  be  readily 
seen. 


22  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

Suppose  we  take  the  range  in  tube  diameters  from  i  inch  to  6  inches, 
and  let  our  pressures  vary  from  100  pounds  to  600  pounds,  the  latter  a 
trifle  above  the  581  pounds  for  which  the  formula  is  supposed  correct. 
Log.  i  is  o  and  log.  6  is  0.778;  log.  100  is  2  and  log.  600  is  2.778.  This 
gives  us  a  range  of  0.778  in  the  value  of  the  logarithm  in  each  case.  Let 
us  make  the  length  of  line  corresponding  to  this  range  the  same  on  the 
two  outside  axes,  say  7.78  divisions  on  whatever  scale  may  be  convenient. 

On  account  of  this  equality  we  may  write  for  the  maximum  values 
of  P  and  D 

I,  (log.  P  -  2)  =  3  19  log.  D 
or 

/!  =  3  12  and  12  =  -. 
o 

Then  l^  (log.  P  —  2)  =  /t  log.  D,  showing  that  the  two  scales  are  iden- 
tical so  far  as  graduation  is  concerned. 

The  logarithms  of  the  values  of  D  will  be  laid  off  from  the  horizontal 
base  line;  the  logarithms  of  P,  above  100,  from  the  same  line.  But  it 
must  be  remembered  that  the  real  zero  for  the  P-line  is  20  divisions 
(on  the  scale  we  have  chosen)  below  the  base  line,  and  that  consequently 
the  line  corresponding  to  A  B  of  Fig.  n  will  slope  up  from  this  point 
to  the  point  marked  i  inch  on  the  .D-line.  There  is  no  need  to  draw  it, 
however. 

The  location  of  the  /-line,  as  given  by  equation  (9),  is 


/,       C  B       i  ' 

Next  let  us  determine  the  modulus  13  for  the  support,  or  axis,  for  t. 
According  to  equation  (10) 

/./.     i^ 

'  h  +1,  ~  4  v 

Inasmuch,  however,  as  the  log.  t  is  multiplied  by  3,  it  will  be  con- 
venient to  consider  its  modulus  as  J  lv  and  graduate  log.  t  directly  with 
this  scale  instead  of  using  a  modulus  of  J  and  laying  off  the  values  of 
3  log.  t.  The  other  quantity  log.  50,210,000  laid  off  on  this  axis  will, 
however,  only  be  affected  by  the  modulus  J  /,  since  the  coefficient 
of  this  logarithm  is  i  instead  of  3. 

In  graduating  the  /-line  note  that  we  must  add  log.  50,210,000  or 
7.7007  to  log.  t.,  and  that  the  zero  from  which  the  graduations  are  meas- 
ured must  be  on  the  sloping  A  5-line  referred  to  above.  If  the  left-hand 
end  of  the  line,  corresponding  to  point  A  ,  is  20  divisions  below  the  hori- 


THE  ALINEMENT   CHART  23 

zontal  base  line,  the  zero  for  the  /-line  will  be  5  of  these  same  divisions 
below.  Now,  using  a  scale  one-fourth  the  size  of  that  used  on  the  outside 
axes  (since  /3  =  i  /J,  lay  up  77.007  divisions.  This  will  give  us  the  point 
corresponding  to  i  inch  on  the  /-line.  Our  values  for  /,  being  less  than  i, 
will  all  fall  below  this. 

For  example,  take  /  =  o.i  inch;  log.  t  =  —  i.  This  will  be  measured 
down  from  point  i  on  the  /-axis,  the  length  being  30  divisions  on  the  one- 
fourth  scale  or  10  divisions  on  the  three-fourths  scale;  or,  what  is  the 
same  thing,  we  may  go  up  47.007  divisions  from  the  zero.  The  other 
points  on  this  axis  may  be  located  in  the  same  way  or  by  joining  up  suit- 
able points  on  the  outside  axes.  The  chart  now  needs  only  to  be  lettered 
to  be  complete. 

A  simple  modification  of  the  alinement  chart  as  already  described 
is  sometimes  of  value. 

Let  our  general  equation  have  the  form  au+  bv=o. 

In  this  equation  c  has  been  made  zero,  and,  since  this  is  so,  y  in  equa- 
tion (7)  is  also  zero.  This  shows  that  the  support  for  the  points  of  inter- 
section is  now  the  line  A  B.  In  order  to  have  the  points  of  intersection 
lie  between  the  points  A  and  B  it  will  be  necessary  that  Au  and  Bv  axes 
lie  on  opposite  sides  of  the  line  A  B.  As  indicated  in  the  last  example, 
there  is  no  necessity  that  A  B  should  lie  perpendicular  to  the  axes,  and  it 
will  evidently  be  to  our  advantage  to  make  it  sloping,  since  in  this  way 
the  chart  can  be  made  to  occupy  less  room. 

CHART  FOR  TWISTING  MOMENT  OF  A  SHAFT. 

The  methods  followed  in  constructing  this  diagram  will  be  shown  by 
working  out  another  practical  example.  For  this  purpose  let  us  take 
the  equation  for  the  twisting  moment  in  a  cylindrical  shaft 

M  =  o.i96£>3/, 
or 

-0.196  D*f+M  =  o, 

where  M  is  the  twisting  moment,  D  the  diameter  of  the  shaft  in  inches, 
and /the  fiber  stress  in  pounds  per  square  inch. 
Let 

u  =  —  /j  /  and   v  =  12  M7 
then 

u  v 

f= —  and  M  =  — - — -, 


CONSTRUCTION    OF   GRAPHICAL   CHARTS 


and 


o.ig6D3u          v 


or 


•.196  D*  12  u  + 

.    /o\ 


=  0 


=  o 


0.19  2  u        !  v  =  o 
Now  from  equation  (8) 

b-a  ^-0.196  D*  12 

b+a  +  0.196  D3  12 


15000-9 

FIG.  14. — Alinement  chart  for  the  twisting  moment  in  cylindrical  shafts. 

This  is  the  equation  for  graduating  the  support  for  D.  The  two 
axes  must  be  graduated  according  to  the  equations  u  =  —  /x/,  and 
v  =  l2M,  which  show  that  the  divisions  on  each  axis  are  to  be  equal  among 
themselves,  or  that  the  graduation  is  regular.  Let  us  assume  that  the 
greatest  fiber  stress  we  shall  need  is  15,000  pounds  and  that  our  largest 
shaft  will  be  4  inches  in  diameter.  Our  maximum  moment  will  then 
be  about  188,200.  If  we  make  i  inch  equal  to  1000  pounds  on 


THE  ALINEMENT  CHART  25 

this  axis  will  have  to  be  15  inches  long.  Making  i  inch  equal  to  10,000 
pounds  on  the  moment  axis  will  give  us  a  length  of  about  19  inches;  l± 
will,  therefore,  equal  10  12.  Suppose  we  say  that  20  inches  will  be  a 
convenient  length  for  the  diagonal,  then  d  will  equal  10  inches. 

Now  graduate  the  outside  axes  into  inches  and  tenths,  taking  as  the 
zero  point  on  each  the  intersection  of  the  axis  and  the  diagonal.  The 
graduations  for  the  .D-axis  or  diagonal  will  be  determined  by  solving 
our  equation  for  x.  Let  us  find  the  point  corresponding  to  the  4-inch 
diameter.  From  equation  (11) 

10  —  0.196X64X1 

#«=io =  —i. ii. 

10+0.196X64X1 

The  division  mark  for  the  4-inch  diameter  will,  therefore,  be  placed 
1. 1 1  inches  to  the  left  of  the  middle  of  the  diagonal.  As  many  other 
points  as  may  be  considered  necessary  are  found  and  laid  off  in  the  same 
manner.  In  Fig.  14  this  has  been  done  for  every  quarter  inch  from 
i  inch  to  4  inches.  To  save  work,  the  graduations  on  the  fiber  stress  line 
need  not  have  been  extended  below,  say,  8000.  The  line  on  which  the 
diameters  are  laid  off  need  not  extend  beyond  the  4-inch  graduation, 
but  for  the  sake  of  clearness  it  has  been  retained  here. 


IV 


in 


in 


DOUBLED  OR  FOLDED  SCALES. 

When  an  alinement  chart  is  intended  to  cover  a  considerable  range  of 
values  we  are  confronted  with  the  difficulty  that  it  must  be  large,  and 
therefore  awkward  to  handle,  or  we  must  have  scale  divisions  which  are  too 
small  for  accurate  reading.  These 
difficulties  may  be  overcome  with 
but  little  additional  trouble  by  a 
system  of  double  graduation  of 
the  axes. 

In  Fig.  15  let  A  and  C  be  the 
outside  axes  of  an  alinement  chart, 
and  B  the  support  on  which  the 
results  are  to  be  read.  Say  we 
wish  to  graduate  the  A  -axis  for  a 
length  equal  to  a-c,  and  that  this 
length  is  too  great  for  our  chart 
if  we  use  a  desirable  scale  unit. 


A  B     B'  c  c' 

FIG.  15. — Diagram  of  an  alinement  chart 
with  doubled  scales. 

Take  a  length  a-b,  equal  to  about 


half  of  a-c  and  lay  this  off  on  the  left-hand  side  of  A  and  graduate  it. 
On  the  right-hand  side  of  A  lay  off  the  rest  of  the  length,  or  b-c.     Call 


26 


CONSTRUCTION   OF   GRAPHICAL   CHARTS 


the  first  scale  I  and  the  second  II.  On  the  C-axis  do  the  same, 
graduating  the  first  half  of  the  desired  length  (which  we  will  call  d-e) 
up  the  left-hand  side  of  the  axis,  and  the  second  half,  or  e-f,  on  the  other 
side.  Mark  them  I  and  II  to  correspond  with  A.  The  location  of  the 
central  support  and  its  scale  unit,  or  modulus,  is  determined  as  previously 


.5- 

8*-i 

8"- 

-8* 

,4- 

3- 

p-3 

^5  Ft.                 5  Ft.  - 

7*- 

7"- 

5                           5 

-7" 

r- 

— 

-17           I 

— 

^-2                            6*7 

r4  Ft               4  Ft.  -E 

-6' 

-14 

-                                • 

•2- 

-13 

—                               — 

—12 

~                                       Z! 

6*- 

-11 

5»_ 

Z       '                     ~ 

-5" 

-10 

I                            z 

3  3  Ft. 

-9            I 

— 

3-8  Ft.               3  Ft.  ^ 

-8         l- 

-1 

_ 

4"- 

-7        .9- 

-.9                           4*- 

—                                 _ 
— 

-4" 

.1- 

—    fi 

-.8 

—                                 . 

_ 

D 

.7- 

-.7 

—                                *. 



I5        .6- 

-.6 

—                                 — 

3"- 

—2  Ft, 

r4     -5- 

-.5                            3"- 

-2Ft.                2Ft.-- 

-3* 

_. 

;3         .4- 

-.4 

- 

~_  Scale  of  Widths 
Scale  of  Areas 

.3- 

\ScaleofAreas 
-.3 

Scale  of  Lengths 

2- 

2"- 

-2" 

_ 

.2- 

-.2 

I 

II 

III 

"iv                               i 

_U        III 

IV                             I 

~n                  iif 

IV 

-IFt. 

-1 

-IFt.                IFt.  - 

.9 

-11" 

.8        .1-* 

-.1 

-  11*                       11*- 

-10* 

.7 

-  10*                      10*- 

.6 

-  9" 

-  9*                         9"- 

-.5 

1"- 

-  8* 

r- 

-  8*                          8*- 

-i' 

.4 

FIG.  16. — An  alinement  area  chart  with  doubled  scales. 

explained  for  the  simple  alinement  chart.  The  left-hand  side  will  be 
graduated  with  values,  say  from  g  to  h,  corresponding  to  I  and  I  on  A 
and  Cy  and  marked  I,  while  the  right-hand  side  will  be  graduated  from 
h  to  i,  corresponding  to  II  and  II  on  A  and  C,  and  marked  II. 

So  long  as  we  wish  to  get  values  on  B  corresponding  to  I  and  I,  or  to 
II  and  II  on  A-  and  C-axes,  we  evidently  have  no  trouble,  but  if  we 


THE  ALINEMENT   CHART  27 

attempt  to  combine  I  on  A  with  II  on  C  we  find  no  place  on  B  where  the 
result  can  be  read.  We  are,  therefore,  compelled  to  use  two  new  axes, 
one  for  values  of  B  and  the  other  for  C.  Call  these  new  axes  B'  and  C'. 
On  Cf  graduate  the  left-hand  side  exactly  the  same  as  the  right-hand 
side  of  C,  or  from  e  to/,  and  the  other  side  like  the  opposite  side  of  C,  or 
from  d  to  e.  Mark  these  scales  III  and  IV,  respectively,  and  since  the 
III  side  of  Cr  is  to  be  combined  with  the  a-b  length  on  A,  the  latter  had 
better  also  be  marked  III.  For  the  same  reason  mark  the  right-hand 
side  of  A,  IV. 

The  central  axis,  B',  must  be  located  in  the  same  relation  to  A  and  Cf 
as  was  B  to  A  and  C,  and  will  be  graduated  on  the  left  to  correspond 
with  the  combination  Ill-Ill  on  A  and  C',  and  on  the  other  side  to  corre- 
spond with  the  combination  IV-IV  on  the  same  axes. 

At  first  sight  this  diagram  is  a  little  confusing  and  there  is  always  a 
chance  for  mistakes  in  connecting  up  wrong  pairs  of  axes.  If  a  little 
care  is  taken,  however,  to  see  that  the  readings  are  made  on  axes  bearing 
the  same  Roman  numeral,  the  seeming  confusion  will  disappear  and  the 
liability  to  error  will  be  small. 

The  process  of  constructing  this  chart  is  so  simple  that  further  expla- 
nation seems  unnecessary.  For  purposes  of  illustration  the  area  chart 
shown  in  Fig.  12  is  reproduced  in  Fig.  16  by  this  method.  A  comparison 
with  Fig.  12  will  show  that  while  the  new  chart  is  somewhat  more  com- 
plex in  appearance,  it  permits  the  use  of  divisions  which  are  so  much 
larger  that  they  compensate  in  a  large  measure  for  the  additional 
confusion. 

ALINEMENT  CHART  WITH  CURVED  SUPPORT. 

All  of  the  alinement  charts  dealt  with  so  far  have  had  straight-line 
axes  or  supports  for  the  different  scales.  This  is  by  no  means  necessary 
since  any  one  or  all  of  them  may  be  curved. 

A  case  in  which  the  intermediate  support  is  curved  will  next  be  con- 
sidered. Let  the  equation  take  the  form 


2 

This  is  the  equation  fo'r  the  space  passed  over  by  a  body  falling 
under  the  influence  of  gravity  and  starting  with  an  initial  velocity.  In 
it  5  =  the  space  moved  over  in  the  time  t,  V  =  the  inital  velocity,  and 
g=the  acceleration  of  gravity,  which  we  will  call  32. 

The  formula  is  chosen  not  so  much  for  its  practical  value  as  because 


CONSTRUCTION  OF  GRAPHICAL  CHARTS 


4100  - 


150  — 


+  200  — 


+  250- 


— f-250 


— 1-200 


—I- 150 


FIG.  17. — An  atonement  chart  with  a  curved  support,  solving  the  equation  S  =  Vt-\ 


THE   ALINEMENT   CHART  2Q 

its  form  is  a  good  one  for  the  purpose  of  illustrating  this  type  of  chart. 
Let  us  write  it 

S=  Vt+  i6t2, 
or 

-Vt  +  S  =  i6t\ 
Make 

u  =  —  /j  F  and  v  =  12S, 
then 


Substituting  above  we  have 

ut       v 

—+r=l6/- 

l\       lt 
or 

I2tu+I1v=i6lll2t2. 

This  is  evidently  identical  with  the  fundamental  equation  for  the 
alinement  chart.     From  equations  (7)  and  (8)  for  y  and  oc  we  have 


J.+1.I 

and 


These  are  the  equations  of  the  points  constituting  the  support  for  /. 

The  choice  of  the  scale  units  is  of  little  or  no  importance  in  this  case, 
since  we  are  not  obliged  to  work  between  any  definite  limits.  For  sim- 
plicity in  calculation,  then,  let  us  take  l^  =  ly 

Then 


and 

i—t 


x  =  d- 


Here  our  formula  does  not  have  a  logarithmic  form,  and  we  can, 
therefore,  graduate  our  scales  in  lengths  proportional  to  the  numerical 
values  of  the  quantities  involved  and  not  of  their  logarithms.  This  has 
been  done  on  the  scales  for  V  and  S.  It  should  be  observed  that  since 
the  modulus  for  F,  or  llt  is  negative,  the  positive  values  of  that  quantity 
are  measured  down  from  the  base  line.  The  distance  between  the  axes 


30  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

may  be  anything  we  like,  but  to  simplify  our  calculations  we  will  make 
it  20  of  some  unit  in  order  that  the  half  distance,  or  d,  may  be  10. 
Solve  the  equation 

i—/ 
x=io 

i  +  t 

for  as  many  values  of  /  as  are  wanted.  In  the  chart  shown  in  Fig.  17  the 
values  taken  for  t  were  o,  1/2,  i,  2,  3,  4,  5,  6,  7,  8,  9,  10,  and  n.  The 
corresponding  values  of  x  are  id,  3.33,  o,  —3.33,  — 5>-6,-7-i5>-7-5> 
—  7.78,  — 8,  — 8.18,  — 8.33.  For  the  same  values  of  t  we  have  for  y:  o,  2.6, 
8,  21.3,  36,  51.1,  66.6,  82.2,  98,  113.8,  129.5,  i45-2>  IOI-4- 

Plot  the  curve  for  these  values  of  x  and  y,  and  letter  it  to  correspond 
with  /.  The  construction  for  the  point  /  =  3  has  been  indicated  by  dotted 
lines. 

The  horizontal  axis  on  which  x  is  plotted  is  only  used  for  the  con- 
struction of  the  curve  and  may  be  omitted  in  the  completed  chart.  It 
is  retained  in  Fig.  17  in  order  that  the  process  may  be  clearly  indicated. 

If  we  connect  two  points  on  the  outside  axes  by  a  straight  line,  the 
intersection  of  this  line  with  the  curved  support  will  give  /,  or  by  connect- 
ing the  initial  velocity  V  with  the  time  on  the  curved  support  we  read  on 
the  5-line  the  distance  passed  over.  This  has  been  done  in  the  figure 
for  F=3O  and  /  =  3,  giving  the  value  for  S  as  234.  By  making  the  index 
line  pass  through  V  =  o  and  the  given  time  we  get  a  case  corresponding 
to  the  simple  law  of  falling  bodies.  If  V  be  taken  negative  we  may  get 
two  intersections  with  the  /-line,  and  either  of  the  times  thus  found  will 
satisfy  the  equation. 


CHAPTER  III. 

ALINEMENT  CHARTS  FOR  MORE  THAN  THREE  VARIABLES 
CHART  FOR  HELICAL  COMPRESSION  SPRING. 

So  far  the  alinement  charts  as  described  have  only  taken  account  of 
three  variables.  This  is  not  a  necessary  limitation  and  we  will  next  con- 
sider a  case  in  which  the  number  of  variables  is  four.  For  illustration 
we  will  use  the  formula  for  the  load  supported  by  a  helical  compression 
spring 

P  =  o. 

where  P  is  the  load,  d  the  diameter  of  the  wire,  r  the  mean  radius  of  the 
coil,  and  /  the  fiber  stress.  Say  we  wish  to  have  our  chart  cover  wire 
from  No.  10  to  No.  oooo  B.  S.  gage,  or  from  0.102  to  0.46  inch  diam- 
eter. Let  us  assume  that  the  mean  radius  of  the  smallest  spring  will  be 
1/2  inch  and  of  the  largest  2  inches,  and  that/  may  vary  between  30,000 
and  80,000  pounds.  Put  the  equation  into  its  logarithmic  form 
log.  P  =  log.  0.196+3  log.  d  +  log.f  -  log.  r. 

We  will  have  to  make  two  steps  in  getting  our  solution,  and  in  each 
step  but  three  variables  must  appear.  Therefore  let  us  say 

log.  0.196  +  3  log.  d  —  log.  r  =  log.  q 
and 

log.  P  =  log.  q  +  log.  f. 

These  two  equations  are  evidently  of  the  same  form  as  those  pre- 
viously treated  by  the  alinement  chart,  and  will  be  charted  by  exactly  the 
same  methods.  The-  quantities  d  and  r,  or  rather  their  logarithms,  we 
will  plot  on  the  outside  axes  and  read  q  on  the  intermediate  support.  See 
Fig.  1 8.  Since  log.  d  and  log.  r  are  affected  by  opposite  signs,  the  positive 
values  of  these  quantities  will  be  laid  off  in  opposite  directions  from  the 
base  line.  As  previously  explained,  the  base  line  may  be  made  sloping, 
and  for  convenience  we  will  suppose  that  this  has  been  done  here.  Our 
former  constructions  depended  upon  a  knowledge  of  the  position  of  this 
line,  but  once  the  matter  is  understood  there  is  no  real  necessity  for  ac- 
tually locating  it,  and  in  the  present  instance  it  will  be  disregarded. 

We  have  assumed  that  the  values  of  r  are  to  lie  between  0.5  inch  and 
2  inches.  The  logarithm  of  0.5  is  —0.301  and  of  2  is  +0.301,  making 
a  total  range  of  0.602.  Choosing  a  suitable  scale  unit,  this  length  is  laid 
off  on  a  vertical  line  at  the  right  of  the  paper.  The  middle  point  will 


CONSTRUCTION   OF   GRAPHICAL   CHARTS 


- No.  0000 


410'- -No.  000 


-  -No.  00 


-  -No.O 


-  -  No.  1 


.258-  -No.  2 


.229 --No.  3 


.204- 


lo.  4 


.182- -No.  5 


-  -  No.  6 


-  -No.  7 


- -No.  8 


.114 --No.  9 


1500  - 


1000 
900 
800 
700 
600 
500 
400 

300  — 


200  — 


40  — 
30  — 

20  — 


-  .6 


-  .7 


-Sprang  Lo&d  in  Lbs.,  P. 

Scale  of  Coil  Radius,  r. 


r,  d  Support 


Scale  of  Wire  Sizes 
Diam.  and  B.  &  S.  No. 


-1.6" 
-1.7" 
-1.8  " 
-1.9" 


-TNo.  10 


FIG.  1 8. — Alinement  chart  for  determining  load  supported  by  a  helical  spring. 


ALINEMENT   CHARTS    FOR    MORE    THAN    THREE    VARIABLES  33 

evidently  be  lettered  i,  and  will  be  the  point  at  which  this  axis  is  inter- 
sected by  the  base  line — a  matter  of  no  importance,  however,  in  the  pres- 
ent instance.  If  we  call  directions  upward  positive  and  downward  nega- 
tive, and  remember  that  log.  r  has  a  minus  sign,  we  will  see  that  the  point 
corresponding  to  2  will  be  at  the  lower  end  of  the  line  and  that  correspond- 
ing to  0.5  at  the  upper  end.  Graduate  the  intermediate  portions  for  as 
many  values  as  are  desired,  of  course,  in  their  logarithms.  The  other 
outside  axis,  on  which  d  is  to  be  laid  off,  is  drawn  to  the  left  of  the  axis 
just  constructed  and  may  be  placed  in  any  convenient  position.  The 
values  of  d  called  for  lie  between  0.102  and  0.46  inch  for  which  the  loga- 
rithms are  1.0086  (or  —0.9914)  and  1.6628  (or  —0.3372).  Log.  d  is  to 
be  multiplied  by  3,  however,  and  therefore  these  values  become  —2.9742 
and  — 1.0116.  Their  difference  is  1.9626  which,  after  multiplication  with 
the  scale  unit,  gives  the  graduated  length  of  the  J-axis.  If  we  use  a  scale 
unit  of  1/3  the  size  of  that  used  on  the  r-line  we  will  get  substantially  the 
same  length  for  the  two  axes.  It  will  be  convenient,  in  graduating  this 
line,  to  take  the  r-line  unit  and  graduate  the  logarithms  of  d  directly 
from  it  rather  than  use  the  i  /3  scale  and  then  multiply  by  three,  since  log.  d 
is  to  be  multiplied  by  3. 

The  position  of  the  zero  on  this  line  (corresponding  to  i)  will  evi- 
dently be  beyond  the  upper  end,  since  all  the  logarithmic  values  are 
negative.  The  point  marked  0.46,  being  nearer  i  than  0.102,  will  be  at 
the  upper  end  and  the  other  at  the  lower.  Having  chosen  the  positions 
for  the  limits  of  this  line,  we  proceed  to  graduate  it. 

The  logarithm  of  0.102  is  1.0086.  Lay  your  engineer's  scale  on  the 
line  so  that  the  point  chosen  to  represent  0.102  is  opposite  0.86  on  the 
scale.  Then  with  the  aid  of  a  table  of  logarithms  pick  off  the  intermedi- 
ate points  up  to  66.28.  Our  formula  shows  that  log.  0.196  should  be 
added  to  3  log.  d.  The  method  of  making  this  addition  was  explained 
in  the  previous  problems  where  we  worked  from  the  base  line.  In  the 
present  case  where  we  are  ignoring  the  exact  position  of  the  base  line  we 
disregard  the  log.  0.196  since  its  only  effect  is  to  change  the  distance  of  our 
indefinite  base  line  from  what  we  must  look  upon  as  the  fixed  position  of 
the  d-line  graduations. 

While  speaking  of  the  J-line  I  should  like  to  call  attention  to  the  re- 
markably regular  appearance  of  the  graduations.  The  nearly  equal  spac- 
ing means  that  the  diameters  of  the  wire  increase  by  approximately  a 
geometrical  progression. 

We  must  next  locate  the  position  of  the  r-d  or  q  support  and  deter* 
mine  the  value  of  its  scale  unit.    The  scale  units  on  the  r-  and  d-  lines  are 
3 


34  CONSTRUCTION    OF   GRAPHICAL   CHARTS 

in  the  ratio  of  i  to  i  /3-  If  we  take  the  unit  for  r  as  the  standard  of  refer- 
ence, we  find  from  equation  (10)  that  the  unit  for  the  r-d  support  will  be 

and  from  the  ratio  of  the  unit  lengths  on  the  outside  axes  we  find  that  the 
intermediate  support  should  divide  the  distance  between  them  in  the  pro- 
portion of  1/4  to  3/4.  Equation  (9.)  This  line  may  now  be  drawn  and 
might  be  graduated  in  the  unit  we  have  determined  if  there  were  any  need 
to  have  the  numerical  result  of  the  r-d  operation.  As  this  will  not  usually 
be  wanted,  we  will  save  ourselves  the  trouble. 

Take  now  the  second  of  the  equations  started  with, 

log.  P  =  log.  q  +  log.  f 

which  shows  that  P  is  the  product  of  the  multiplication  of  q  and/.  Their 
scales  will  be  the  outside  axes  of  a  new  chart  and  P  will  be  graduated  on 
an  axis  between  them.  We  have  assumed  a  variation  of  /  from  30,000  to 
80,000.  The  logarithm  of  30,000  is  4.4771  and  of  80,000  is  4.9031. 
The  difference,  0.4260,  multiplied  by  the  scale  unit  chosen,  gives  the 
length  of  the  axis.  In  the  chart  made  for  this  article  the  scale  unit  selected 
for  the  /-axis  is  1/2  that  of  the  reference  standard  used  on  r.  It  would 
have  been  better  on  some  accounts  if  the  unit  had  been  made  larger  in 
order  to  get  greater  scale  lengths  on  the  different  axes.  I  found,  however, 
that  any  larger  scale  unit  that  I  could  use  would  give  a  unit  for  graduat- 
ing the  P-axis  which  would  be  utterly  impracticable  with  the  ordinary  engi- 
neer's scale.  The  graduation  of  the  P-scale  might,  of  course,  be  made  by 
a  series  of  projections  from  the  other  axes  had  there  been  any  pressing 
need  to  have  the  /-scale  long,  but  this  is  a  tedious  operation.  In  the 
problem  we  are  considering  the  values  of /will  generally  be  expressed  in 
round  numbers,  and  there  will  be  no  need  of  minute  subdivision — the 
chief  advantage  of  a  long  scale.  Accordingly,  the  unit  value  of  1/2 
was  chosen  for/ 

In  locating  the /-line  it  was  simply  placed  as  far  to  the  right  as  it  would 
conveniently  go  without  interfering  with  the  r-line,  and,  as  with  the  other 
axes,  the  graduations  are  located  on  it  in  any  position  we  please.  Begin- 
ning at  the  lower  end,  which  we  mark  30,000,  we  graduate  up  with  the 
logarithms  of  the  desired  fiber  stresses  until  we  reach  80,000. 

Lastly,  we  must  locate  and  graduate  the  P-axis.  The  scale  unit  for 
the  r-d  support  has  been  found  to  be  1/4;  that  of  the /-line  is  1/2. 
Therefore,  substituting  in  formula  (10),  we  get  for  the  scale  unit  for  P 


- 

3    HI 


ALINEMENT   CHARTS    FOR    MORE    THAN    THREE   VARIABLES  35 

The  P-axis  will  divide  the  distance  between  the/-  and  g-lines  into  parts 
which  have  a  ratio  of  2/3  to  1/3,  since  the  units  on  the  side  lines  are  1/2 
and  1/4.  Equation  (9).  The  range  over  which  we  must  suppose  the 
values  of  P  to  vary  is  a  trifle  indefinite.  It  will  not  do  to  substitute  the 
values  of  the  variables  already  settled  upon,  which  give  the  minimum  and 
maximum  values  of  P,  for  this  would  lead  to  absurd  combinations.  It  is 
not  probable,  for  instance,  that  a  spring  would  be  iriade  of  No.  10  wire  and 
a  2 -inch  coil  radius,  and  it  is  still  less  likely  that  wire  0.46  inch  in  diameter 
would  be  used  in  a  spring  whose  coil  radius  was  1/2  inch.  Taking  average 
conditions,  I  find  that  the  range  for  P  should  be  somewhere  in  the  neigh- 
borhood of  from  10  to  1000  pounds.  To  make  sure  of  being  on  the  safe 
side,  I  have  extended  the  limits  a  little  beyond  each  of  these  values. 

Now,  when  it  comes  to  starting  the  graduations  on  P  we  ought,  properly 
speaking,  to  know  the  location  of  the  base  line;  but  we  have  completely 
lost  track  of  this,  and  it  cannot,  therefore,  serve  us.  We  may  easily  locate 
one  point  on  P,  however,  if  we  run  through  a  trial  calculation.  Let 
d=  0.102  inch,  r  =  i  inch,  and/  =  50,000  pounds.  Then 

.     O.I023 

P  =  0.196- —  — 50,000  =  10.4. 

On  the  chart  join  up  d  =  0.102  with  r  =  i,  and  find  the  intersection 
with  the  r-d  support.  From  this  point  draw  a  line  to  50,000  on  the 
/-line  and  find  its  intersection  with  the  line  chosen  for  P.  This  must  be 
the  point  corresponding  to  10.4,  whose  logarithm  is  1.017.  We  have  thus 
found  a  starting  point  for  our  graduations,  and  the  other  marks  may  easily 
be  located  with  the  proper  scale,  i  /6  that  of  r.  The  chart  is  now  com- 
plete except  for  lettering.  For  convenience  in  reading  I  have  given  the 
d-line  a  double  set  of  numbers,  one  for  the  diameters  and  the  other  for 
the  corresponding  gage  numbers. 

To  read  the  chart  draw  a  line  between  the  selected  values  of  r  and  d 
(say  0.8  and  0.204)  and  get  the  intersection  with  the  r-d  support.  Con- 
nect this  point  with  the  chosen  fiber  stress  (say  80,000) .  The  intersection 
with  P}  which  is  at  166,  gives  the  load  the  spring  will  carry. 


CHART  FOR  STRENGTH  OF  GEAR  TEETH. 

Next,  let  us  take  a  formula  containing  five  variables  instead  of  four. 
The  principles  involved  are  precisely  the  same  as  those  already  discussed: 
we  merely  carry  the  process  one  step  further.  For  the  sake  of  variety 
a  slight  change  will  be  made  in  the  disposition  of  the  axes.  The  formula 


36  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

chosen  for  charting  is  the  well-known  one  by  Lewis  for  the  strength  of 
gear  teeth 

W  =  s  pfy 

where  W  is  the  pitch-line  load,  5  the  fiber  stress,  p  the  circular  pitch,  / 
the  face  width,  and  y  a  constant  corresponding  to  the  number  of  teeth. 

Let  us  separate  the  right-hand  side  of  the  equation  into  two  parts  and 
construct  a  separate  chart  for  each,  one  giving  the  product  of  s  and  y, 
and  the  other  the  product  of  p  and/.  Then,  if  we  take  the  resulting  prod- 
uct lines  as  the  outside  lines  of  a  new  chart,  we  will  find  the  value  of  W 
(their  product)  on  their  intermediate  support.  See  Fig.  19. 

We  must  impose  the  customary  limits  on  the  variables  in  order  to 
determine  the  size  of  the  chart.  Suppose  we  let  p  vary  between  i  /2  and 
2  inches  and  /  between  i  and  6  inches.  According  to  the  tables  which 
usually  accompany  the  formula,  s  may  vary  between  1700  and  20,000 
and  y  from  0.067  f°r  a  i2-tooth  pinion  to  0.124  for  the  rack.  For  the 
sake  of  simplicity  we  will  suppose  the  application  of  the  chart  to  be 
limited  to  the  15 -degree  involute  teeth. 

Take  first  the  values  of  y.  The  logarithm  of  0.067  is  2.8261,  and  of 
0.124  it  is  1.0934,  giving  a  difference  between  the  extremes  of  0.2673; 
this  multiplied  by  the  scale  unit  chosen  gives  the  graduated  length  of  the 
axis.  Pick  out  the  values  of  y  from  the  table,  find  their  logarithms,  and 
lay  down  the  latter  on  the  axis,  making  the  lower  end  of  the  line  the  loga- 
rithm of  0.067.  Lewis  gives  a  formula 

0.684 

y  =  o.i24  —  — — 
n 

for  calculating  the  value  of  y  from  the  number  of  teeth.  I  have  made  these 
calculations  and  laid  off  the  results  on  the  other  side  of  the  line  for  purposes 
of  comparison.  It  will  be  noted  that  the  tabular  values  are  spaced  some- 
what irregularly  as  compared  with  the  calculated.  This  is  a  matter  of 
passing  interest,  but  the  chief  point  to  which  I  wish  to  direct  attention  is 
the  ease  with  which  empirical  constants,  which  are  connected  by  no  known 
law,  may  be  handled  by  these  diagrams.  There  is  no  need  of  trying  to 
force  them  to  fit  some  arbitrary  equation  for  they  may  be  inserted  in  the 
chart  exactly  as  they  were  obtained  from  experiment.  In  lettering  this 
line  we  place  opposite  the  graduations  the  numbers  of  teeth  corresponding 
to  the  different  values  of  y,  which  we  have  plotted,  instead  of  the  ^-number 
themselves.  The  former  we  know  from  our  given  gear,  while  the  latter 
is  of  no  special  interest.  This  line  from  now  on  will  be  called  the  n-  instead 
of  the  y-axis. 

Opposite  and  parallel  to  this  line  we  draw  the  axis  for  s,  the  fiber  stress. 


ALINEMENT   CHARTS    FOR    MORE    THAN    THREE   VARIABLES 


37 


The  logarithm  for  its  lowest  value,  1700,  is  3.2305,  and  for  the  highest, 
20,000,  is  4.3010,  giving  a  difference  of  1.0705.  If  we  take  a  scale-unit 
value  of  one-fourth  that  used  on  the  w-axis  the  two  lines  will  be  approxi- 


mately  equal.  Taking  the  lower  end  of  the  line  at  any  convenient  point, 
mark  it  1700  and  graduate  up  to  the  top  in  the  logarithms  of  the  desired 
values  of  s.  In  lettering  this  line  it  might  be  well,  in  case  the  gears  for  which 


38  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

the  chart  is  to  be  used  are  all  to  be  of  the  same  material,  to  place  opposite 
the  fiber  stresses  the  appropriate  speeds  as  shown  by  the  table,  thus  making 
the  chart  entirely  self-contained.  Where  several  different  materials  are 
to  be  used  this  would  probably  cause  a  considerable  amount  of  confusion, 
and  it  has  therefore  been  omitted  here. 

The  lengths  of  the  scale  units  on  the  outside  axes  being  i  and  i  /4,  we 
find  the  unit  length  for  use  on  the  intermediate  support  to  be 


and  the  support  will  divide  the  distance  between  the  outside  axes  into 
intervals  whose  lengths  are  1/5  and  4/5  of  this  distance.  We  do  not 
graduate  the  intermediate  support,  since  the  numerical  results  of  the 
multiplication  are  not  wanted. 

Next  take  the  values  of  p  and  /;  p  varies  from  i  /2  inch  to  2  inches. 
The  corresponding  logarithms  are  —  0.301  and  +0.301,  making  a  total 
range  of  0.602.  The  lowest  value  of  /is  i  inch  (log.  =  6)  and  the  highest 
6  inches  (/#£.  =  0.778),  making  a  total  range  of  0.778.  Since  these  two 
lengths  are  so  nearly  equal  we  might  as  well  use  the  same  scale  unit  for 
each,  and  it  will  be  found  convenient  to  make  it  i  /5  that  used  on  the  w-axis. 
The  support  for  the  product  will  have  a  scale  unit  1/2  the  size  of  that 
used  on  the  outside  axes,  or  will  be  equal  to  i  /io  the  length  of  that  which 
was  used  on  the  w-line.  This  support  and  the  one  previously  located 
are  to  be  used  as  the  outside  axes  for  the  last  multiplication,  whose  product 
is  W.  The  size  of  the  scale  unit  on  the  PF-line,  since  those  on  its  outside 
axes  are  i  /io  and  i  /5,  is 


and  the  line  itself  will  divide  the  distance  between  these  axes  in  the  ratio  of 
1/3  to  2/3. 

It  will  be  convenient  to  have  the  TF-line  fall  between  the  diagrams  used 
for  the  preliminary  multiplications  in  order  to  avoid  confusion.  There- 
fore, locate  it  somewhat  to  the  right  of  the  w-axis  and  then  draw  a  vertical 
for  the  support  for  the  p-f  product  so  that  its  distance  from  the  TF-line  is 
i  /  2  the  distance  of  the  latter  from  the  n-s  support.  At  convenient  equal 
distances  from  the  />-/  support  draw  the  p  and  /-axes,  and  graduate  them 
with  the  logarithms  of  p  and/,  using  a  scale  unit  i  /5  the  size  of  that  we 
used  for  the  w-graduation.  As  before,  we  may  locate  the  graduated  parts 
of  these  lines  anywhere  we  please  on  them. 

The  TF-axis  is  now  to  be  graduated,  and  its  graduations,  unlike 
the  others,  must  start  at  some  definite  point.  Solve  the  equation  for 


ALINEMENT  CHARTS  FOR  MORE  THAN  THREE  VARIABLES     39 

any  values  within  the  prescribed  limits.     Take,   for  instance,  n=i2, 
y  =  0.067,  S=I 7°°j /=  I y  and  p=i/2.     Then 

w  =  1700  x  1 1 2  x  i  x  0.067 =  56.95. 

On  the  chart  join  1700  on  the  s-line  with  12  on  w,  and  get  the  inter- 
section with  the  intermediate  axis,  which  will  be  at  the  product  (unknown) 
of  the  two.  Join  i  on  the /-line  with  0.5  on  the  />-line,  and  get  the  inter- 
section with  their  intermediate  axis,  giving  again  the  product  (unknown) . 
Join  the  the  product  of  n  and  s  with  that  of  p  and/,  and  the  intersection 
with  the  JF-line  must  be  the  point  corresponding  to  56.95.  Its  logarithm 
is  1.7555.  Lay  an  engineer's  scale  with  the  proper- sized  graduations 
(1/15  that  used  on  the  w-line)  on  the  PF-line  so  that  1.7555  on  it  is  at tne 
point  we  have  located,  and  graduate  the  rest  of  the  line  from  a  table  of 
logarithms.  The  method  of  using  the  chart  should  be  obvious  from  what 
has  preceded,  but  may  be  briefly  recapitulated.  Join  the  desired  values 
on  n  and  s,  say,  27  and  10,000,  by  a  straight  line  and  mark  its  intersec- 
tion with  the  support.  Join  the  desired  values  of  p  and/,  say,  i  and  3, 
by  a  straight  line  and  get  its  intersection  with  their  support.  Join  these 
two  points  by  a  third  line,  and  its  intersection  with  the  TF-line  gives 
the  load,  3000  pounds,  which  the  gear  will  carry  safely. 

I  believe  that  a  comparison  of  this  diagram  with  others  which  have 
been  published  for  the  solution  of  this  equation  will  show  that  it  has  some 
very  marked  advantages  over  them  in  point  of  clearness  of  reading  and 
simplicity  of  construction.  The  only  point  which  gave  any  trouble  in 
construction  was  the  selection  of  scale  values  for  the  different  lines  so 
that  they  might  all  be  read  from  an  ordinary  engineer's  scale.  Several 
trials  were  necessary  before  they  were  finally  settled. 

Enough  has  been  said,  I  think,  to  indicate  the  general  method  to  be 
followed  in  cases  where  the  equation  to  be  charted  contains  more  than 
three  variables,  and  there  should  be  no  difficulty  in  extending  the  method 
to  any  case  where  more  than  five — the  largest  number  treated  here — are 
involved.  Before  leaving  this  part  of  the  subject,  however,  I  wish  to  take 
up  briefly  another  case,  differing  slightly  from  those  which  have  gone 
before,  and  which  is  occasionally  serviceable  in  special  problems. 

CHART  FOR  STRENGTH  OF  A  RECTANGULAR  BEAM. 

Suppose  we  have  an  equation  of  the  form 

WL_bh2 

~8~       6  '" 
This  is  the  equation  for  a  rectangular  beam,  supported  at  the  ends 


40  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

and  uniformly  loaded.  In  it  W  is  the  total  load,  L  the  length  of  the  beam 
in  inches,  b  the  breadth,  and  h  the  height  of  the  rectangular,  cross- 
section  of  the  beam,  both  in  inches,  and  /the  fiber  stress. 

Let  us  suppose  for  convenience  that  the  beam  is  of  white  oak  or  long- 
leaf  yellow  pine  for  which  the  "  Cambria  )J  pocket  book  gives  a  safe  fiber 
stress  of  1200.  Our  formula  may  then  be  simplified  to  read 

WL=i6oobh\ 

For  our  limits  let  us  say  thatL  varies  between  10  and  24  feet,  or  120 
and  288  inches,  b  from  2  to  10  inches,  and  h  from  4  to  14  inches.  Then 
W  will  vary  from  about  178  to  26,100.  Suppose,  now,  we  construct 
two  charts,  one  for  multiplying  W  and  I,  and  the  other  for  1600  b  times 
h2.  Fig.  20.  The  two  products  are  to  be  equal.  We  may,  therefore, 
use  the  same  line  as  the  support  for  the  product  for  each  chart  if  the 
scale  units  on  the  two  supports  have  the  same  value.  The  base  lines  for 
the  two  charts  may  or  may  not  coincide,  but  it  is  essential  that  they  inter- 
sect the  intermediate  support  at  the  same  point  if  we  expect  the  two  index 
lines  to  cut  it  at  a  common  point.  This  must  be  the  case  if  the  products 
of  the  two  multiplications  are  to  be  equal  as  we  have  supposed.  As  in 
the  previous  illustrations,  there  is  no  necessity  for  actually  drawing  the 
base  line.  The  general  method  of  procedure  in  constructing  this  diagram 
is  so  similar  to  what  has  gone  before  that  it  will  not  be  described  in  detail. 

After  finding  the  range  of  values  required  for  the  L-line  we  choose  a 
convenient  unit  length  and  graduate  the  line  in  the  logarithms  of  the 
desired  values.  The  TF-line  is  placed  opposite  it  at  any  convenient 
distance  and  graduated  with  a  scale  unit  whose  length  is  one-quarter 
that  used  on  L.  The  support  for  the  product  of  these  quantities  must, 
therefore,  divide  the  distance  between  them  in  the  ratio  of  i  /5  to  4/5,  and 
its  scale  unit  will  be 


For  the  b-  and  /j-lines  it  will  be  found  that  a  scale  unit  of  the  same  size 
as  the  standard  used  on  L  may  be  taken  for  6,  and  one  of  one-quarter  the 
standard  for  h.  These  will  give  convenient  lengths  for  the  two  axes,  and 
the  intermediate  axis  will  also  have  a  scale  unit  of  i  /5,  since  again 


This  is  essential  if,  as  remarked  above,  the  products  of  the  two  multi- 
plications are  to  be  represented  by  equal  lengths  on  the  common  support. 
Remember  that  when  h2  is  plotted  the  lengths  of  the  logarithms  of  h 
must  be  multiplied  by  2.  The  scale  units  chosen  for  the  b-  and  &-lines 


ALINEMENT    CHARTS    FOR    MORE    THAN    THREE    VARIABLES 


being  i  and  1/4,  the  support  must  be  distant  from  these  lines  in  the  ratio 
of  4/5  to  1/5.  Lay  off  the  b-  and  /Klines  at  any  convenient  distances 
from  the  W-L  support  which  will  satisfy  this  ratio. 


10 


8"- 


r- 


3  - 


24' 


14' 
13" 
12" 
11" 
10" 


9 

8" 

-7" 


Scale  of  Height,  h. 


Scale  of  Length,  L. 


"Common  Support 
for  all  Scales. 


F24' 

E-23' 
1.22' 

=-21' 

»' 

19' 

=-18' 

-17' 
16' 
-15' 
•14' 
13' 
12' 


10 


Scale  of  Load.  W. 


FIG.  20. — An  alinement  chart  plotted  from  the  equation  W  L  =  i6oo  bh2. 

Graduate  the  Mine  in  the  logarithms  from  2  to  10  with  a  scale  unit  of  i. 
The  h-\me  is  to  be  graduated  in  twice  the  logarithms  of  the  numbers 
between  4  and  14.  The  position  of  the  graduations  on  b  is  chosen  arbi- 


42  CONSTRUCTION   OF   GRAPHICAL   CHARTS 

trarily,  but  for  h  must  be  found  by  a  simple  trial  calculation,  since  the 
location  of  the  base  line  is  unknown. 

Assume  6  =  2  inches,  &=  5  inches,  and  L  =  160  inches  (13  feet  4  inches). 
Then 


1  60 

Join  500  on  the  TF-line  with  160  inches  (13  feet  4  inches)  on  theZ-line 
and  mark  the  intersection  with  the  intermediate  axis.  Through  this 
point  of  intersection  draw  another  line  so  as  to  pass  through  2  on  the  6-line. 
Where  this  line  intersects  the  h-lme  must  be  the  point  numbered  5.  Its 
logarithm  is  0.699.  Knowing  this  and  the  proper  scale  length,  we  may 
easily  find  the  other  points  on  this  line.  To  read  the  chart  draw  a  line 
between  the  chosen  values  of  W  and  L  and  mark  its  intersection  with 
the  support  for  the  product.  Any  lh|e  drawn  through  this  point  to  the  b- 
and  &-axes  will  intersect  them  in  values  which  will  give  the  necessary 
strength  to  the  beam.  Thus,  on  the  chart,  the  solution  has  been  found 
for  the  case  where  W=$2OO  and  L  =  200  inches  (16  feet  8  inches).  It  is 
found  that  a  beam  4  x  10  inches  will  satisfy  the  conditions  as  to  strength. 


CHAPTER  IV. 
THE  HEXAGONAL  INDEX  CHART. 

A  type  of  chart  of  quite  a  different  character  from  any  of  those  pre- 
viously described  will  now  be  considered.  Suppose  we  have  a  diagram 
like  Fig.  21,  where  A  O  C  is  any  angle  whatever,  and  OB  its  bisector. 
Measure  equal  distances  Oa  and  Oc  on  the  OA-  and  OC-axes  and  erect 
perpendiculars  ab  and  cb.  They  meet,  of  course,  on  O  B.  The  length 
Oa  =  Oc  =  Ob  cos.  AOB,  or  Oa+Oc  =  2  Ob  cos.  A  O  B. 

Now  suppose  b  moved  out  to  V  on  the  perpendicular  b  b'.  Project 
b'  to  of  and  c1 '.  Then  since  b  b1  makes  the  same  angles  with  O  A  and 
O  C,  its  projections  on  these  two  axes  will  be  of  equal  length,  or  a  a'  will 
equal  c  c' .  Therefore, 

Oa+Oc  =  Oa'+Ocf  =  2  Ob  cos.  A  OB. 

We  have  here,  evidently,  a  new  form  of  addition  chart.  If  the  scale 
values  on  O  A  and  O  C  are  equal,  and  that  on  O  B  is  this  unit  times 

— ,  the  length  O  b  measured  to  this  unit  is  equal  to  the  sum  of 
2  cos  A  OB 

Oa'  and  Oc'.     If  A  O  C  is  90  degrees  the  unit  length  for  the  O  J5-axis  is 

that  used  on  O  A  multiplied  by— -=,  and  if  AO  C  is  120  degrees  the  unit 

V2 
lengths  on  all  three  axes  are  the  same. 

If  we  were  to  graduate  the  three  axes  with  their  proper  units  and  then 
erect  perpendiculars  to  the  axes  at  the  division  points  we  could  find  the 
sum  of  Oa'  and  Oc'  by  finding  the  perpendicular  from  O  B  which  passes 
through  the  point  of  intersection  of  the  perpendiculars  from  a'  and  c'. 
It  will  readily  be  seen,  however,  that  this  would  entail  a  very  confusing  net- 
work of  lines,  and  it  is,  therefore,  customary  with  this  form  of  chart  to 
use  what  is  known  as  a  transparent  index.  It  consists  of  a  transparent 
sheet,  preferably  of  thin  celluloid,  on  the  lower  side  of  which  are  ruled 
three  lines  meeting  at  a  point;  each  line  is  perpendicular  to  one  of  the  axes. 
The  axes  having  been  properly  graduated,  the  index  is  laid  on  the  chart 
(care  being  taken  that  the  index  lines  are  perpendicular  to  their  respective 
axes)  and  is  so  adjusted  that  one  perpendicular  passes  through  the  se- 
lected value  on  O  A  and  the  second  through  that  on  O  C.  The  third 

43 


44 


CONSTRUCTION  OF  GRAPHICAL  CHARTS 


perpendicular  will  then  intersect  O  B  at  the  sum  of  the  two  quantities. 
The  angle  A  O  C  may  be  anything  we  like,  but  since  we  get  equal  scale 
units  on  the  three  axes  with  an  angle  of  120  degrees,  it  is  advantageous, 
in  general,  to  use  that  value.  Where  this  is  done  the  arrangement  is 
known  as  the  "hexagonal"  type.  The  whole  thing  is  so  simple  and  self- 
c 


FIG.  21. 


FIG.  22. 


a'       b' 

FIG.  23.  FIG.  24. 

Diagrams  illustrating  the  hexagonal  index  chart. 

evident  that  it  scarcely  seems  to  call  for  an  illustrative  example,  and  I  will, 
therefore,  not  attempt  to  do  more  than  refer  to  some  of  its  more  important 
peculiarities. 

Like  the  other  forms  of  addition  chart  already  examined,  it  may  be 
turned  into  a  chart  for  multiplication  by  graduating  the  axes  in  the  log- 
arithms of  the  numbers  instead  of  the  number  themselves,  Fig.  22. 
When  the  graduation  of  the  middle  or  O5-axis  is  identical  in  general 
form  (not  necessarily  in  length)  with  those  on  the  side  axes,  it  may  be 


THE  HEXAGONAL   INDEX   CHART  45 

projected  from  them  by  parallel  lines  whose  angle  with  O  A  or  O  C  is 
the  supplement  of  the  angle  which  O  B  makes  with  them.  This  is  a  con- 
venience in  case  the  angle  O  A  C  is,  say,  90  degrees,  as  it  does  away  with 

the  necessity  for  a  scale  whose  unit  length  is  — 7=  times  that  used  on  O  A 

V2 

and  O  C.  It  will  also  be  noted  by  reference  to  Fig.  23  that  the  gradu- 
ated lengths  on  the  three  axes  may  be  moved  as  far  as  we  please  in  a 
direction  perpendicular  to  these  axes  without  changing  the  points  at 
which  the  index  line  cuts  them.  This  is  sometimes  an  advantage  in 
that  it  allows  us  to  shift  our  scales  so  as  to  get  a  more  compact  and  con- 
venient arrangement  of  the  chart  than  is  always  possible  if  the  axes  are  to 
meet  at  O.  For  instance,  suppose  we  wished  to  arrange  the  three  scales 
on  the  sides  of  an  equilateral  triangle,  shown  dotted  in  Fig.  23.  It  is 
plain  that  we  get  precisely  the  same  results  with  the  lines  a'  &',  c'  d' ',  and 
e'f  that  we  do  with  the  lines  a  b,  c  d,  and  ef;  i.e.,  if  a  b+ef=c  d  it  is  like- 
wise true  that  a'  b'  +  e'  f'  =  c'  d'.  It  is  also  advantageous  in  case  any  of 
the  quantities  is  affected  by  a  number  of  different  coefficients.  In  this 
case  it  is  only  necessary  to  draw  a  separate  parallel  line  for  each  value 
of  the  coefficient  multiplied  into  the  variable,  and  graduate  it  with  the 
product  of  the  two.  Then,  taking  the  line  affected  by  the  desired  coeffi- 
cient, pick  out  the  required  point  on  it  and  pass  the  index  line  through 
this  point. 

This  form  of  chart  may  be  arranged  easily  to  take  care  of  a  larger 
number  of  variables  than  three.  On  Fig.  24  the  product  of  O  A  and  O  C 
will  be  found  on  O  B.  If  we  draw  a  new  axis  O  D,  making  an  angle  of 
1 20  degrees  with  O  B,  we  have  a  new  diagram  on  which  we  may  obtain 
the  product  of  O  B  and  O  D.  The  product  will  be  read  on  O  C,  or  any 
line,  as  O  E,  parallel  to  it.  This  operation  may  be  repeated  an  indefinite 
number  of  times,  and  it  is  here  that  the  advantage  of  being  able  to  move 
the  scales  in  a  direction  perpendicular  to  themselves  becomes  most 
apparent.  It  enables  us  to  handle  a  large  number  of  variables  and  have 
a  separate  scale  for  each  one  of  them. 

In  problems  of  this  sort  it  is  an  advantage  to  have  a  transparent  index 
made  in  the  shape  shown  in  the  same  figure.  It  is  a  hexagon  with  the 
sides  parallel  to  the  index  lines.  This  chart  takes  its  name  from  the  shape 
of  this  index  sheet.  After  setting  the  index  to  get  the  product  on  O  B, 
place  a  straight-edge  against  the  side  parallel  to  the  O  B  index  line,  and 
it  is  easy  to  slide  it  into  position  for  the  next  reading  without  losing  its 
orientation,  and  at  the  same  time  always  keep  the  index  through  the 
point  last  found  on  O  B. 


46  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

A  MODIFICATION  OF  THE  PRECEDING  TYPE. 

Personally,  I  must  confess,  the  method  of  the  transparent  index  does 
not  appeal  to  me  very  strongly.  It  has  the  disadvantage  of  not  being 
self-contained,  and  unless  we  provide  a  special  index  for  each  chart  the 
two  are  not  likely  to  be  found  together  when  they  are  wanted.  In  the 

second  place,  it  is  easier  to  "  fudge," 
or  force  the  index  to  give  the  desired 
results  than  with  most  of  the  other 
types.  Still  it  must  be  admitted  that 
it  has  its  advantages  in  certain  cases, 
and  I  have  had  one  or  two  problems 
— A  to  chart  which  it  seemed  impossi- 

FIG.  25.— Diagram  illustrating  a  modification     ble   to   handle   with   any    approach 
of  the  preceding  type.  ,.   .       , 

to  simplicity  by  any  other  method. 

A  form  of  chart  which  is  related  to  both  the  hexagonal  and  alinement 
types  is  shown  in  Fig.  25  In  it  the  axes  O  A  and  O  C  make  any 
angle,  and  O  B  bisects  it.  Draw  any  line  a  c.  Then  from  similar  tri- 
angles we  have 

Od  _Oc-bd 
Oa  '       Oc 
or 

Od  bd 


Oa  Oc 

and 

Od       bd 

. .     I     —     •»• 

Now 

Od=bd=         °b 


2  cos.  A  O  B 
i         2  cos.  A  OB 


Oa       Oc        c      Ob 

The  simplest  case  is  where  the  angle  A  O  B  is  60  degrees;  then  cos. 
A  OB  =  i/2  and 


iii 


Oa       Oc       Ob 

This  is  in  reality  the  "reciprocal"  form  of  the  type  just  described. 
The  equation  we  have  derived  is  of  the  same  form  as  that  which  was 
used  in  plotting  the  chart  shown  in  Fig.  7, 

JL     I     _f_ 

/       /'       p  '  j  nvj  Lijijol  JuJ  3njo«.j 


THE  HEXAGONAL   INDEX   CHART 


47 


As  a  matter  of  interest,  this  formula  has  been  recharted  by  the  new 
method.  In  Fig.  26  /  and  /'  are  graduated  on  the  outside  axes  and  p 
on  the  middle.  To  read  the  chart  join  up  points  on  two  of  the  axes  which 
are  known  and  get  the  intersection  of  the  line  with  the  third.  This  will 
be  the  value  necessary  to  satisfy  the  equation. 

The  6o-degree  arrangement  of  the  axes  is  not  quite  so  satisfactory 
in  this  type  of  chart  as  in  the  last  de- 
scribed, since  when  we  are  working 
out  toward  the  limits,  the  index 
Kne  is  likely  to  cut  some  one  of  the 
axes  at  an  angle  which  is  disagree- 
ably acute.  For  this  reason  it  is 
generally  considered  that  the  ad- 
vantage lies  with  a  smaller  angle 

even  if  the  work  of  graduating    is 

somewhat  more  difficult. 

Where  the  two  outside  axes  are  graduated  alike,  the  central  axis  may 
be  marked  off  without  much  difficulty  by  simply  joining  like  points  on 
the  outsides.  The  marks  thus  found  on  the  middle  axis  will  have  num- 
bers whose  values  are  one-half  those  on  the  outside  lines.  This  form  of 
chart  might  be  used  for  multiplication  by  plotting  the  reciprocals  of  the 
logarithms  of  the  numbers  to  be  multiplied  on  the  outside  lines  and  of 
their  products  on  the  middle.  The  advantages  of  'such  an  arrangement 
are  not  very  apparent,  however,  and  it  has  but  little  practical  interest. 


^.-^hart  plotted  by^method  illustrated 


CHAPTER  V. 
PROPORTIONAL  CHARTS. 

A  family  of  chart-forms  of  great  structural  simplicity  is  that  which  is 
known  under  the  general  name  of  the  "proportional"  or  "parallel  aline- 
ment"  type.  The  ease  with  which  they  may  be  laid  out  and  the  fact  that 
they  may  be  used  with  certain  forms  of  equations  which  cannot  be  handled 
so  conveniently  by  those  types  previously  described  are  strong  recom- 
mendations for  their  use  in  these  cases. 

Take  any  two  lines  meeting  at  any  angle  and  lay  off  the  distance  a  and 
6,  as  shown  in  Fig.  27.  Connect  the  points  at  the  ends  of  these  lengths 
by  a  straight  line  and  draw  a  parallel  to  it.  This  parallel  intersects  the 
axes  at  the  lengths  c  and  d.  From  similar  triangles  we  have 


I 


If,  therefore,  we  lay  off  on  one  side  of  the  vertical  axis  a  scale  for  the 

values  of  a,  and  on  the  other  side  for  c,  and  similarly,  on  the  horizontal 

,  _  axis,  the  scales  for  b  and  d,  we  have 

**  -  £j  -  »H 

a  chart  which  takes  account  of  four 
variables.  Knowing  a,  &,  and  c,  for 
instance,  we  join  a  and  b  by  a 
straight  line  and  draw  a  parallel  to 
it  through  c.  This  line  intersects 
the  d  axis  at  the  required  value  of 
that  variable. 

It  may  be  advantageous,  in  certain 
cases,  to  have  the  scales  graduated 
on  separate  lines  instead  of  doubling 
up  as  was  done  with  a  and  c  or  b  and  d.  This  is  also  shown  in  Fig.  27 
where  two  lines  parallel  to  the  original  axes  have  been  drawn.  The  solu- 
tion d!  is  found  by  drawing  through  cr  a  parallel  to  the  original  a  b  line. 

CHART  FOR  STRENGTH  OF  THICK  HOLLOW  CYLINDERS. 

As  an  illustration  of  this  type  of  chart  take  the  Lame  formula  for  the 
strength  of  thick  hollow  cylinders  subjected  to  internal  pressure 


o« — 


.-d— 


FIG.  27. — Diagram  of  proportional  chart. 


where  D  is  the  outside  diameter  of  the  cylinder,  d  the  inside  diameter 

48 


PROPORTIONAL   CHARTS 


49 


(both  in  inches),  /the  fiber  stress  in  the  material,  and  p  the  internal  pres- 
sure (both  in  pounds  per  square  inch). 

Squaring  both  sides  of  the  equation  we  have 


"  f-p' 

This  has  the  same  form  as  the  fundamental  equation.  Plot  on  the 
horizontal  and  vertical  axes  the  desired  values  of  D2  and  d2.  On  the 
same  axes  plot  as  many  values  oif+p  and/—  p  as  may  be  deemed  neces- 


is- 


14,-1<XXX> 


-9000 
18- 

8000 


^--TOOO^ 


I 

-6000  I  Key-         (dwithD 

W  Connect   Jf_pwfthf+p 

.2    10*-k5000 


.4000 


7- 
6.,-2000 

S- 


Scate^fqrf+p 

i  .  i  >^i  .7  .  r>o  .^,1.^ 


(NOoVj-iO   CO     V-      00       0>          O         *^l  Vj  CO  -^«  "io  \0  t-  1)0 

•^•^r^rH»HfH  ?-«  ?•*  r^ 

Scale  for  External  Diameter  D. 
FIG.  28. — Proportional  chart  for  the  strength  of  thick  hollow  cylinders. 

sary.  The  scale  units  used  for  corresponding  quantities  on  the  two 
axes  may  be  equal  or  not,  as  we  please.  In  this  case  if  we  use  equal  scale 
units  the  horizontal  axis  will  be  considerably  longer  than  the  other,  and 
the  index  lines  are  likely  to  cut  it  at  a  disagreeably  acute  angle.  Accord- 
ingly the  values  of  D  and/+/>  are  laid  off  with  a  scale  unit  whose  length 

4 


50  CONSTRUCTION    OF   GRAPHICAL   CHARTS 

is  2/3  that  used  for  d  and/—  p.     On  the  chart  the  solution  is  shown  for 
/=8ooo,  p  =  40oo  and  d=io  inches,  giving  ^  =  17.3  inches. 

The  only  objection  which  might  be  raised  against  the  chart  just  shown 
is  the  fact  that  a  preliminary  calculation — the  addition  and  subtraction 
of  the  quantities  /  and  p — is  necessary  before  the  chart  is  used.  This, 
however,  is  not  the  fault  of  the  chart  but  of  the  equation  which  was  pur- 
posely chosen  to  bring  up  this  point.  A  makeshift  of  this  sort  should,  of 
course,  be  avoided  where  possible,  but  is  often  not  objectionable.  In  this 
case  where  the  values  of /and  p  will  usually  be  given  in  round  numbers 
the  necessary  computations  otf+p  smdf—p  are  easily  made  mentally 
and  no  serious  difficulty  will  result.  I  have,  however,  seen  this  scheme 
used  on  some  charts  where  it  involved  quite  a  little  calculation  or  con- 
sultation of  tables  and  where,  on  account  of  the  complexity  of  the  equa- 
tion, it  was  evidently  the  only  method  which  permitted  it  to  be  charted 
at  all. 


THE  ROTATED  PROPORTIONAL  CHART. 

This  type  of  chart  is  susceptible  of  a  slightly  different  arrangement 
which  is  sometimes  considered  advantageous.  Suppose  the  lines  carrying 
the  quantities  c  and  d,  Fig.  27,  to  have  been  rotated  about  the  origin,  O, 
through  90  degrees.  We  will  have  a  diagram  like  Fig.  29.  In  making 
this  rotation  the  line  joining  the  points  c  and  d  will  likewise  turn  through 

90  degrees,  and  will  be  at  right  angles 
instead  of  parallel  to  that  joining  a  and  b. 
In  reading  such  a  chart  it  is  generally 
customary  to  have  a  transparent  index 
consisting  of  a  sheet  of  thin  celluloid  with 
two  lines,  at  right  angles  to  each  other, 
scratched  on  its  lower  surface.  This  is 
laid  on  the  chart  in  such  a  way  as  to  have 
one  of  the  lines  pass  through  a  and  b  and 
the  other  through  c.  The  intersection  of 
the  latter  with  the  d  scale  then  gives  the 
required  value  of  that  quantity.  The  same  result  may  be  obtained,  of 
course,  by  a  pair  of  draftsman's  triangles  laid  against  each  other. 

With  this  chart,  as  with  the  first  one  described,  there  is  no  need  that 
the  axes  carrying  a  and  d,  or  b  and  c  should  coincide.  Every  condition 
will  be  satisfied  if  the  lines  are  separate  but  parallel.  The  advantage  of 
this  arrangement  of  chart  over  the  other  is  not  very  marked,  and  I  do  not 


FIG.  29. — Diagram  of  rotated  pro- 
portional chart. 


PROPORTIONAL   CHARTS  51 

incline  much  toward  its  use.  Some  authorities,  however,  seem  to  look 
upon  it  with  considerable  favor  and  that  is  my  main  reason  for  referring  to 
it  at  all. 

CHART  FOR  RESISTANCE  OF  EARTH  TO  COMPRESSION. 

One  formula  will  be  worked  out  showing  its  application.  For  this 
purpose  let  us  take  the  formula  for  the  resistance  of  earth  to  compression, 
used  in  calculations  for  foundations.  It  is: 

„          7  /  i  +  sin. 

P  =  wh  ( ! : — 

\  i  —  sin. 

where  P  is  the  ultimate  load  on  the  earth  in  pounds  per  square  foot,  w  is  the 
weight  of  the  earth  in  pounds  per  cubic  foot,  h  is  the  depth  in  feet  and 
<£  the  angle  of  repose  of  the  earth. 
The  expression 

-\-sin. 


i  —  sn. 

may  be  treated  as  a  single  variable  and  the  equation  arranged 

P  w 


h       /i  — sin.  <A2. 
\i  +  sin.  (/)/ 


This  gives  us  the  simple  proportion  we  need  for  this  type  of  chart. 
The  limits  were  determined  as  follows:  The  friction  angles  given  by 
Rankine  for  different  conditions  lie,  roughly,  between  15  and.  45  degrees, 
though  they  exceed  this  in  a  few  cases.  To  cover  them  all  the  gradua- 
tions on  the  $  scale  will  be  run  up  to  60  degrees,  though  it  is  probable  that 
most  of  the  values  wanted  will  lie  below  40  degrees.  The  extreme  value 
of  h  was  arbitrarily  taken  as  1  5  feet.  The  values  of  w  given  in  the  pocket- 
books  range  from  about  70  to  130  pounds.  Taking  h  as  15  feet,  w  as 
130  pounds,  and  </>  as  40  degrees,  we  find  P  to  be  about  40,000. 

Next  let  us  choose  our  scale  units.  If  we  take  the  scale  unit  for  h 
(which  we  will  call  /t)  as  1/4,  then  15X1/4=3  3/4  inches,  which  is  about 
the  length  wanted  in  the  original  drawing.  For  w  let  the  scale  unit  (74)  be 
taken  as  -£$.  Then  I3OX  i  /4O=3  i  /4  inches,  again  a  convenient  length. 
For  the  ^>-axis  let  the  unit  length  (/3)  be  7^4  The  maximum  value  of 
the  parenthesis  containing  (f>  is  0.347  when  <j)=  15  degrees.  Then  0.347  X 
•OT=  8-675  inches,  which  will  be  about  right. 

Now  the  scale  units  should  be  in  the  same  ratio  as  the  quantities  they 
affect.  Hence,  calling  the  scale  unit  for  P  12  we  have 


Q.04 


CONSTRUCTION   OF   GRAPHICAL  CHARTS 


Then 


v->  — 


004 


'60 


Multiplying  the  maximum  value  of  P  by  this  unit  we  get. 
40,000  X-jTnnr=I°  inches 


15  - 

-Ov>UW 

16  - 

• 

17  - 

-30000 

18 

19 

-      o: 

-250003* 

20  - 

g 

*«•- 

1 

|  a  " 

1 

*8  23  - 

-20000  "I 

J|  24  ' 

1 

£ey"        jwwithp                        25N 
Connect  }Pwith£ 

26  • 

27  - 

-150QO              ,• 

*\       f 
\    * 

28 

'  '         s'  ^ 

29 

','*****?  \ 

_..  _ 

'                           \ 

>' 

\ 

^ 

-10000                        \ 
\ 

,   •  .                                             \ 

* 

\ 

/             34 

s 

36 
38 
X                                     40  - 
s 
45- 
50- 

1?: 

\ 
\ 

V 

\ 
\ 

-  5000 

\ 

•  ,                                                                       \ 
\ 

N 
\ 
\ 

\ 

nTA£«.UA.       ,    ~*  <  /"|,,u;rt  T?rt^*  ^^  tr*««*-u 

Depth,  h. 

in  Pounds. 
PlG.  30. — Proportional  chart  for  earth  resistance  in  compression. 


PROPORTIONAL   CHARTS  53 

as  the  length  of  the  P  axis.  This  was  a  trifle  greater  than  I  wanted  for 
the  limits  I  had  placed  on  the  size  of  the  chart  and  I  arbitrarily  reduced 
it  to  about  the  same  length  as  the  0-line,  making  the  maximum  value  for 
P,  35,000.  This  would  correspond  to  an  angle  <£  of  about  38  degrees 
with  h  and  w  at  their  maximum,  and  would  probably  cover  most  cases. 
It  was,  however,  entirely  a  matter  of  convenience  and  there  is  no  reason, 
in  a  practical  case,  why  the  scale  should  not  extend  as  much  further  as 
the  conditions  in  the  problems  likely  to  be  encountered  would  seem  to 
require.  The  graduation  of  the  different  scales  is  now  an  easy  matter 
and  the  completed  diagram  is  shown  in  Fig.  30. 

The  broken  lines  show  the  position  of  the  index  forw=i2O,  ^=30 
degrees,  and  ^=15  feet,  giving  the  load  P=  16,200  This  is,  as  stated 
above,  the  ultimate  strength  of  the  soil.  If  a  fixed  factor  of  safety  may 
be  used  for  all  cases  likely  to  be  met  with,  it  might  easily  be  introduced 
when  P  is  plotted;  that  is,  the  numbers  placed  opposite  the  graduation 
marks  on  this  scale  would  be  divided  by  whatever  factor  we  chose.  Then 
the  diagram  would  give  us  safe,  instead  of  ultimate  loads. 

CHARTS  WITH  PARALLEL  AXES  FOR  SUMS  OR  DIFFERENCES. 

Next  let  us  take  a  case  like  that  shown  in  Fig.  31.  Here  the  quan- 
tities are  laid  off  from  an  arbitrary  zero  line  on  two  axes  which  are 
parallel.  Draw  a  transversal  between  the  ends  of  the  lengths  a  and  b, 
and  another  parallel  to  it  cutting  the  axes  in  the  lengths  c  and  d.  An  in- 
spection of  the  diagram  shows  that 

a  —  b  =  c  —  dj 

or  the  difference  between  the  lengths  on  the  two  axes  cut  by  any  system 
of  parallel  lines  is  constant.  If  one  pair  of  corresponding  quantities  had 
been  laid  off  below  the  zero  and  the  other  above  it  we  should  have  had 
a  constant  sum  instead  of  a  constant  difference.  We  may  even  get  a  case 
corresponding  to 

a  —  b  =  c  +  d, 

if  we  lay  off  the  quantity  d  below  the  zero  and  the  others  above.  This 
will  be  referred  to  later.  As  in  the  previous  type  of  chart,  there  is  no 
need  to  have  the  values  a  and  c,  or  b  and  d  laid  off  on  the  same  axes. 
They  may  be  laid  off  on  parallel  axes  if  the  distance  between  each  pair  of 
axes  is  the  same.  This  distance  might  be  varied  if  the  need  for  it  arose, 
but  it  would  require  an  alteration  in  the  scale  units  to  correspond. 

To  save  referring  to  it  again,  it  might  as  well  be  noted  here,  once  for 
all,  that  this  chart,  and  all  of  those  yet  to  be  described  involving  four 


54 


CONSTRUCTION  OF  GRAPHICAL  CHARTS 


variables,  has  the  same  rotational  property  as  was  indicated  in  Fig.  29 
for  the  first  type.  This  is  shown  in  Fig.  32,  where  the  c  and  d  axes  have 
been  turned  through  90  degrees  without  altering  their  relative  positions. 
The  position  of  the  index  is  shown  by  the  fine  lines,  and  the  construction 
is  sufficiently  clear,  I  think,  to  render  any  further  explanation  unnecessary. 

c »/ 


FIG.  31.  FIG.  32. 

Diagrams  of  proportional  charts. 

A  chart  of  the  kind  we  have  been  examining  is  of  little  importance  if 
we  are  only  to  use  it  for  addition  and  subtraction;  but  it  acquires  an 
added  value  if,  instead  of  plotting  the  numbers  themselves  on  the  axes, 
we  plot  their  logarithms.  This  transforms  the  chart  into  one  for  multi- 
plication and  division. 

CHART  FOR  CENTRIFUGAL  FORCE. 

An  example  of  the  use  of  it  is  given  in  Fig.  33.  The  formula  used  is 
that  for  centrifugal  force, 


C  = 


w  v 


g 


where  C  is  the  centrifugal  force  in  pounds,  w  is  the  weight  in  pounds,  g 
is  the  acceleration  of  gravity,  v  is  the  velocity  in  feet  per  second,  and  r  is 
the  radius  in  feet  of  the  path  of  the  weight.  Rewrite  the  equation 


w       gr 
Then 

log.  C  —  log.  w  =  2  log.  v  —  log.  gr 

which  is  identical  with  the  fundamental  equation  given  above.  The 
limits  between  which  we  are  to  work  are  of  no  special  importance  here, 
since  the  chart  is  not  supposed  to  be  applied  to  any  particular  problem. 
We  will  have  to  fix  some  conditions,  however,  so  let  us  say  that  w  varies 


PROPORTIONAL   CHARTS 


55 


from  i  pound  to  100  pounds,  v  from  i  foot  to  50  feet,  and  r  from  o.i  foot 
to  10  feet.  The  maximum  value  of  C  will  be  776.4,  and  of  its  logarithm 
2.89.  The  maximum  value  of  log.  v2  is  3.398,  of  log.  w  is  2,  and  of  log. 
32.2  r  is  2.508. 


700- 


500- 
400-1- 


200- 


O  100-1-10 

'T 


I50 

340- 


20- 


-•  5 


4--2 


2- 


100 
90 
80 
70 
CO 
50 
40- 


-.1 


FIG.  33. — Proportional  chart  for  centrifugal  force. 

In  graduating  the  axes  the  same  scale  unit  must  be  used  throughout. 
All  except  the  r-scale  commence  with  mark  i  at  the  zero  point,  and  are 
laid  off  in  any  convenient  sized  divisions  from  a  table  of  logarithms. 
The  C-scale  was  extended  to  800  instead  of  stopping  at  its  exact  upper 


56  CONSTRUCTION   OF   GRAPHICAL   CHARTS 

limit,  776.4.  In  the  case  of  the  r-scale  we  must  place  mark  i  at  a  distance 
of  1.508  (log.  32.2)  above  the  zero,  and  graduate  above  and  below  this  as 
desired.  Fig.  33  shows  the  completed  diagram  with  transversals  drawn 
to  indicate  a  solution  for  w  =  30,  v  =  9,  and  r  =  10;  C  should  then 
equal  7.55. 

CHART  FOR  PISTON-ROD  DIAMETER. 

With  this  type  of  chart  it  is  not  necessary  that  the  equation  be  in  the 
form  of  a  simple  proportion,  though  it  should  be  capable  of  being  placed 
in  that  form  by  a  little  manipulation.  For  instance,  take  the  formula 
given  by  Kent  for  the  diameter  of  the  piston  rod  of  a  steam  engine, 

d  =  0.013  \/  Dip, 

where  d  is  the  diameter  of  the  rod,  D  is  the  piston  diameter,  and  /  the 
length  of  the  stroke,  all  in  inches,  and  p  the  maximum  steam  pressure  in 
pounds  per  square  inch.  Squaring,  this  becomes 

d2  =  0.000169  D  I  p 


5917 
In  its  proportional  form  it  is: 

d* 

7T 

or  in  logarithms, 

2  log.  d  -  log.  D  =  log.  I  -  (log.  5917  -  log.  p), 
which  agrees  with  the  fundamental  equation  for  this  type  of  chart. 

Here  we  plot  the  logarithms  of  d2,  D  and  /  as  usual.  In  the  case  of 
p,  however,  we  first  lay  off  the  log.  of  5917  (  =  3.772)  from  the  zero  and 
from  that  point  plot  the  logarithms  of  p  downward,  since  we  use  the  recip- 
rocal of  p  and  not  p  itself  in  the  last  member  of  the  proportion.  For  the 
sake  of  compactness  it  is  well  to  have  all  four  scales  on  about  the  same 
horizontal  zone,  and  since  those  of  /  and  p  are  much  higher  than  the  others 
we  drop  their  zeros  by  equal  distances  below  those  of  d  and  D.  The 
zeros  are  not  shown  in  the  chart,  Fig.  34,  since  none  of  the  graduations 
go  down  that  far,  and  only  the  working  parts  of  the  scales  are  needed. 
No  error  is  introduced  by  this  shifting  of  the  scales  since  the  slopes  of  the 
lines  joining  them  are  the  same  before  and  after  the  transfer.  The  units 
used  in  graduation  must  be  the  same  for  all  scales  unless  a  different  one 
is  indicated  by  the  exponent  of  the  quantity.  Therefore,  D,  I  and  p  are 
plotted  with  one  unit,  and  d  (since  it  is  squared)  with  one  twice  as  large. 
The  resulting  chart  is  shown  in  Fig.  34,  and  the  broken  parallel  lines  give 


PROPORTIONAL   CHARTS 


57 


the  solution  for  the  case  where  D  =  20  inches,  /  =  30  inches,  and  p  = 
100  pounds.     Then  d  =  3.18  inches. 

It  may  be  mentioned  here  that  this  type  of  chart  may  be  applied  to 
equations  containing  but  three  variables.  If,  for  instance,  in  our  equa- 
tion 

a       c 
~b=~d3 

6"- 


5"  ~ 


4."  - 


3.5"- 


2.5- 


2"  -  -40r 


30" 


1.5- 


24" 


15- 


12  J 


CQ 

-SO' •§ 


15" 
-12" 


FIG.  34. — Proportional  chart  to  determine  piston-rod  diameter. 


a,  b  and  d  are  variables  and  c  a  constant,  the  c  graduation  is  reduced  to 
a  single  point  through  which  all  lines  referring  to  c  must  pass.  The 
method  of  using  such  a  chart  is  precisely  the  same  as  for  those  just  de- 
scribed, and  it  is  hardly  of  sufficient  importance  to  merit  more  than  a 
passing  notice. 


58  CONSTRUCTION   OF   GRAPHICAL   CHARTS 

THE  Z-CHART. 

The  examples  which  have  been  given  will  illustrate  sufficiently  well, 
I  think,  the  general  methods  to  be  followed  in  cases  involving  a  simple 
proportion,  and  we  will  now  proceed  to  examine  a  new  type  which, 
while  it  bears  a  family  resemblance  to  some  of  those  previously  described, 
differs  from  them  in  several  important  particulars. 

In  Fig.  35  we  have  three  axes  arranged  in  the  form  of  a  letter  Z. 
Draw  a  transversal  across  them.  From  similar  triangles  we  have 

a        b  b  d 

—  = — ,  or  a  =  — . 

d        c  c 

Add  d  to  each  side  of  the  equation  and  we  have 

-(b  +  c). 


a+ 


b  d 

---  h  d  = 
c 


FIG.  35. 

Diagrams  illustrating  Z-charts. 

Now  (b  +  c)  ,  the  length  of  the  diagonal  of  the  Z,  is  a  constant  which 
we  may  call  k. 


.-.  a  +  d  =  —k. 
c 

Draw  a  second  line  parallel  to  the  first  transversal.     Then 


(12) 


and  the  original  equation  becomes 


(13) 


If  then  the  equation  which  we  are  to  chart  has  the  form 

x 


PROPORTIONAL    CHARTS  59 

we  lay  off  u  on  the  upper  horizontal,  v  on  the  lower,  oc  also  on  the  lower 
and  y  on  the  diagonal.  Joining  the  values  of  u  and  v  corresponding  to 
a  and  d  by  a  transversal,  and  drawing  a  parallel  to  it  through  the  value 
of  x  corresponding  to  e  we  get  the  resulting  value  of  y,  or  /,  on  the 
diagonal. 

Similarly,  we  may  get  the  solution  of  a  problem  where  the  difference 
of  two  quantities  is  used  instead  of  their  sum.  Fig.  36  shows  the  arrange- 
ment. Here,  as  before, 

a        b  b  d 


—  =  —  or  a  =     — , 
d        c  c 


and 


,      bd       .       d  .        d 

a  —  d  =  --  d  =  —  (b  —  c)  =  —  k 
c  c  c 

.'.  a  —  d  —  —  rk. 

The  selection  of  the  scale  units  is  of  some  importance  with  this  chart 
and  a  brief  discussion  of  their  mutual  relation  is  necessary.  It  is  under- 
stood, of  course,  that  the  numerical  values  of  the  quantities  u,  v,  oc  and  y 
are  to  be  multiplied  by  certain  scale  units  in  order  to  get  their  measured 
lengths,  a,  d,  e  and  /  on  the  axes.  Let  these  lengths  be  lv  12,  13  and  /4 
for  u,  v,  x  and  y,  respectively.  For  u  and  v  the  scale  unit  must  be  the 
same  (^  =  1^),  since  otherwise  parallel  lines  joining  their  scales  would 
not  indicate  a  constant  sum,  but  /3  and  14  may  be  chosen  at  will.  Now 
since  a  =  l1u,  d=l1vj  e  =  l3oc  and/=/4;y  we  have  by  substitution  in 
equation  (13) 


or  snce 


oc 
u  +  v  =—  , 

y 


and 


which  gives  the  necessary  length  of  the  diagonal  of  the  Z. 
For  the  subtraction  formula  this  becomes 


(IS) 


60  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

Sometimes  we  wish  to  make  /3  =  lr     In  this  case 

(b+c)  =  14,  (16) 

or  the  diagonal  is  the  same  length  as  the  scale  unit  used  in  graduating  it. 

It  should  be  noted  here  that  the  a  and  d  scales  may  be  shifted  along 
their  axes  the  same  amount  and  in  the  same  direction  as  far  as  we  please, 
without  changing  the  direction  of  the  transversal  joining  them  and  that, 
therefore,  no  error  will  be  introduced.  This  sometimes  permits  us  to 
make  a  more  convenient  arrangement  of  the  scales  as  will  be  shown  later 
in  connection  with  the  chart  for  chimney  draft. 

CHART  FOR  POLAR  MOMENT  OF  INERTIA. 

As  the  first  illustration  for  the  construction  of  the  Z-chart  I  have 
chosen  the  formula  for  the  polar  moment  of  inertia  of  a  flat  rectangular 
plate  about  an  axis  perpendicular  to  its  plane  and  passing  through  the 
center.  It  is  sometimes  used  in  the  power  calculations  for  the  draw  spans 
of  bridges,  the  assumption  being  that  the  span  may  be  taken  as  having 
approximately  the  same  polar  moment  of  inertia  as  the  flat  plate.  The 
formula  is: 

7=     21(5*+Z*), 

where  /  is  the  polar  moment  of  inertia,  W  the  weight  of  the  plate,  and 
B  and  L  its  breadth  and  length.  The  weight  will  be  expressed  in  pounds 
and  B  and  L  in  feet.  The  engineer  who  wishes  to  have  the  forces  in  his 
final  results  in  pounds  instead  of  poundals,  will  usually  prefer  to  divide 
at  once  by  g,  instead  of  doing  this  at  the  end  of  his  calculations;  in  which 
case  the  formula  becomes 


This  may  be  written 


386.4 
and  we  evidently  have  an  equation  suited  to  the  Z-type  of  chart. 

When  planning  this  chart  my  intention  was  to  give  it  something  like 
a  practical  form  by  taking  the  maximum  values  of  B  and  L  as  about  10 
and  80.  However,  since  these  quantities  are  to  be  squared  and  L2  = 
6400,  while  B2  is  only  100,  it  is  evident  that  if  we  lay  them  off  to  the 
same  scale  and  use  any  practicable  length  for  6400,  100  would  be  so  small 
(after  the  necessary  reduction  by  the  engraver)  that  its  subdivisions  for 


PROPORTIONAL  CHARTS 


61 


o-i«r 


CQ 


90000-35' 

.  37. — Z-chart  plotted  from  formula  for  polar  moment  of  inertia. 


62  CONSTRUCTION    OF   GRAPHICAL  CHARTS 

smaller  values  of  B  would  be  illegible  in  the  cut.  In  view  of  the  fact, 
however,  that  the  charts  which  illustrate  this  book  are  intended  pri- 
marily as  examples  of  methods  of  construction  and  application,  I 
have  not  hesitated  in  many  cases  to  sacrifice  a  practical  chart  for  the  sake 
of  getting  one  which  showed  a  process  clearly,  and  this  I  shall  do  in  the 
present  instance.  The  conditions  are  assumed  to  give  clear  reading  scales 
in  the  cut,  but  the  chart  in  its  present  form  will  have  little  practical  value 
for  the  bridge  designer. 

Let  us  say  then  that  the  maximum  value  for  W  is  to  be  35,000,  that 
the  maximum  value  for  B  is  10  feet,  and  for!,,  30  feet.  Then  the  maxi- 
mum value  for  /'  is  very  nearly  90,000.  The  scale  units  and  scale  lengths 
must  next  be  fixed.  I  wished  to  keep  the  original  drawing  inside  of  a 
length  of  10  inches.  By  making  the  scale  unit  for  /'  ro-UT(T  I  Set  a  gradu- 
ated length  of 

90,000  X  -nro-uir  =  9  inches. 

This  is  the  scale  unit  we  called  /3  in  the  preliminary  explanation.     For 

386.4 


the  PF-line  it  will  be  convenient  to  make  the  scale  unit,  /4,  equal  to 

=  7  inches 


5000 
Then 

386.4 


5000 

is  the  graduated  length  of  this  axis.  This  unit  makes  it  possible  to  plot 
W  directly  from  the  50  side  to  an  engineer's  scale  without  bothering  about 

the  coefficient .     For  the  L  and  B  scales  let  us  take  l±  =  i  /ioo. 

386.4 

Then  for  L2  we  have  a  graduated  length  of  900  X  i  /ioo  =  9  inches,  the 
same  as  for  /',  and  for  B2  ioo  X  i  /ioo  =  i  inch.  Substituting  the  sscale 
units  thus  found  in  equation  (14)  we  get  for  the  length  of  the  diagonal  of 
the  Z, 

i     w  386.4 

,,  ,          h  h          TOTr  5000  3»864,000  Q   •       I, 

(b  -f  c)  =  J  -  = =  —        —  =  7.728  inches. 

l*  Timnr  5°°>°°° 

Having  drawn  our  axes  (the  diagonal  making  any  convenient  angle 
with  the  parallels)  we  have  only  to  graduate  them,  and  this  is  a  simple 
matter,  B  and  L  being  plotted  in  the  squares  of  the  desired  values  with  a 
scale  unit  of  i  /ioo,  while  the  W-  and  /'-lines  are  plotted  directly  from 
the  Winr  and  10000  scales.  The  parallel  broken  lines  show  how  the 
chart  is  read  for  the  case  where  B  =  8,  L  =  20,  W  =  30,000,  which  gives 
for  /'  36,000. 


PROPORTIONAL   CHARTS 


CHART  FOR  INTENSITY  OF  CHIMNEY  DRAFT. 

The  next  formula  which  I  have  charted  is  one  for  the  intensity  of 
chimney  draft 

'7-64       7-95 


f        LI' 

\Tl'      T, 

where/  is  the  draft  expressed  in  inches  of  water,  h  the  height  of  the  chim- 
ney in  feet,  7\  the  absolute  temperature  of  the  chimney  gases,  and  T2 
the  absolute  temperature  of  the  external  air. 

The  formula  will  be  seen  at  once  to  belong  to  the  second  type  of  Z- 
chart  where  we  have  a  difference  instead  of  a  sum  of  two  variables.  The 
general  method  of  procedure  is 
identical  with  that  just  described, 
but  there  are  a  few  differences  of 
minor  detail  which  require  a  brief 
description.  Thus  the  variables  Tl 
and  T2  appear  in  the  denominators 
of  the  fractions  instead  of  the 
numerators,  which  indicates  that 
the  plotted  values  are  proportional 
to  the  reciprocals  of  these  quantities 
and  not  to  the  quantities  them- 
selves. For  our  limits  let  us  take 
h  as  varying  between  50  and  150 
feet,  Tl  from  761  to  1161  degrees 
absolute  (or  from  300  degrees 
Fahrenheit  to  700  degrees  Fahren- 
heit), and  T2  from  461  to  561 
degrees  absolute  (or  from  o  degrees 


FIG.  38. — Z-chart  to  determine  intensity 
of  chimney  draft. 


Fahrenheit  to  100  degrees  Fahren- 
heit). Then  /  will  have  a  maxi- 
mum value  of  1.46  inches  and  we  will  graduate  it  from  o  to  1.5  inches. 
The  scale  unit  on  the  /-line  (/3)  I  took  as  ~  which  gives  a  length  of 
1.5  X  "oTg-  =  7-5  inches  for  its  graduations;  /4,  the  /j-scale  unit  was  taken 
as  i  /4O.  This  gave  a  graduated  length  from  zero  of  150  X  i  /4O  =  3.75 
inches;  llt  the  unit  used  for  7\  and  T2  was  made  i  /o.ooi.  The  extreme 
length  of  the  J'.j-line  from  its  zero  will  then  be 

7.64 


461 


X 


0.00  1 


=  1 6. 6  inches, 


64  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

too  great  for  the  size  of  chart  planned  which  I  wished  to  keep  within  a 
length  of  10  inches.     The  lower  limit  of  the  graduations  on  this  line  is 

X=1.62  inches 


from  the  zero.  This  is  an  empty  space  which  is  of  no  advantage,  and  the 
chart  will  be  improved  in  appearance  and  compactness  if  we  slip  the 
graduations  along  the  axis  toward  the  zero  point  or  the  point  where  the 
diagonal  intersects  this  axis.  In  my  drawing  the  graduations  were 
shifted  a  distance  of  8  inches,  which  brought  them  within  the  prescribed 
limits.  The  7\-graduations  were  shifted  the  same  amount  in  the  same 
direction,  and  thus  no  change  was  made  in  the  direction  of  the  trans- 
versals joining  them  and  no  error  introduced. 

The  length  of  the  diagonal,  from  equation  (15)  is 


(6-e)  =  -j-±          ^-      »,t^  =  S  inches. 

As  many  values  of  -J— -  and  -^—  as  are  wanted  are  now  calculated 

and  plotted,  remembering  that  their  zeros  are  8  inches  beyond  the  points 
where  the  diagonal  intersects  their  axes,  h  is  plotted  on  the  diagonal 
and /on  the  same  axis  with  T2. 

In  lettering  the  7\-  and  jT2-lines  it  will  be  a  convenience  for  the  person 
who  uses  the  chart  to  have  the  temperatures  marked  in  the  Fahrenheit 
scale  instead  of  from  the  absolute  zero.  This  has  been  done  on  the  chart, 
Fig.  38,  but  the  absolute  temperatures  have  been  retained  at  each  end  of 
the  scales  as  an  aid  to  a  clearer  understanding  of  the  construction.  The 
position  of  the  parallel  index  lines  shows  the  application  of  the  chart  to 
the  case  where  the  chimney  temperature  is  400  degrees  Fahrenheit,  the 
temperature  of  the  outside  air  60  degrees  Fahrenheit,  and  the  height  of 
the  chimney  100  feet.  The  draft  gage  reading  should  then  be  a  trifle 
over  0.54  inch. 

CHART  FOR  SAFE  LOAD  ON  HOLLOW  CAST-IRON  COLUMNS. 

An  interesting  application  of  the  Z-type  of  chart  is  to  certain  equations 
where  a  variable  appears  twice.  Since  each  time  it  appears  it  occupies 
a  scale,  the  number  of  variables  we  can  handle  is  reduced  from  four  to 
three.  Suppose  the  fundamental  equation  to  be  of  the  form 

v 

u+v  =  — . 

y 


PROPORTIONAL   CHARTS 


FIG.  39. — Z-chart  to  determine  safe  load  of  hollow  cast-iron  columns. 


66  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

This  evidently  refers  to  equation  (12)  used  in  demonstrating  the 
Z-chart.  Here  one  index  line  instead  of  two  is  used  in  making  a  reading. 

This  gives  a  particularly  useful  chart  since  equations  of  this  type  are 
by  no  means  uncommon  and  are  awkward  things  to  handle  by  any  of  the 
methods  hitherto  described.  A  formula  which  will  serve  as  an  excellent 
illustration  is  the  one  given  below  which  is  taken  from  the  "Cambria" 
pocket  book: 

5 


800  D2 

It  is  the  formula  for  the  safe  load  on  hollow  round  cast-iron  columns 
with  flat  ends.  In  it  P  =  the  safe  load  in  tons  (of  2000  pounds)  per 
square  inch  of  column  section,  D  is  the  outside  diameter  of  the  column 
in  inches  and  L  the  length  of  the  column  also  in  inches.  The  successive 
steps  required  to  put  the  formula  into  working  shape  are  indicated  below 

L2  5 

h  800  D2  ~~      P 
and 


0.2  P 

Let  us  take  the  limits  for  D  as  6  and  15  inches,  for!,  as  72  inches  (  =  6 
feet),  and  288  inches  (  =  24  feet).  Then  P  will  vary  between  1.3  tons  and 
4.86  tons.  The  maximum  value  to  be  laid  off  on  the  L-line  will  be  2882 
or  82,944,  on  the  D-lme  8ooX  I52=  180,000,  and  on  P,  0.2X4.86  =  0.972. 
The  scale  units  for  L  and  D  being  the  same,  it  is  evident  that  the  value 
180,000  will  control  the  choice  of  the  scale  unit  if  we  are  planning  a  chart 
of  a  certain  size.  Suppose  the  scale  unit  /t  is  made  1/20000.  Then 

180,000X1/20000=9  inches, 
which  is  about  right.     For  Z,  we  have 

82,944X1/20000=4.147  inches. 

The  scale  unit  /4  used  in  graduating  P  on  the  diagonal  will  be  taken  as 
i /o.i  and  the  graduated  length  will,  therefore,  be 

0.972X1/0.1  =  9.72  inches. 

Since  /3  =  ^  the  length  of  the  diagonal  intercepted  between  the  parallel 
axes  is,  according  to  equation  (16),  /4  or  i  /o.i  =  10  inches.  On  the  chart, 
Fig.  39,  D2  has  been  graduated  for  every  inch  between  6  and  15  inches, 
and  from  its  zero  the  diagonal,  10  inches  long,  has  been  drawn  in  any 


PROPORTIONAL   CHARTS  67 

convenient  direction.  On  it  we  may  consider  that  we  are  graduating 
0.2  P  with  a  scale  unit  of  i/o.i  or  P  with  a  scale  unit  of  1/0.5.  Lastly, 
the  second  parallel  is  drawn  from  the  end  of  the  diagonal  and  graduated 
for!,2.  This  has  been  done  for  every  12  inches  and  the  points  marked 
with  the  corresponding  values  in  feet. 

As  noted  above,  but  a  single  transversal  or  index  line  is  required  for 
reading  this  chart. 

By  joining  15  feet  on  theL-line  with  10  inches  on  the  ZMine  we  find 
the  safe  load  per  square  inch  on  the  column  is  3.56  tons. 

Before  leaving  this  subject  it  might  be  well  to  call  attention  to  the  fact 
that  another  way  of  treating  three  variables  by  the  Z-chart  is  to  imagine 
that  one  of  the  four  which  normally  belong  to  it  is  replaced  by  a  constant. 
The  scale  belonging  to  it  then  reduces  to  a  point  through  which  all  of  the 
lines  pertaining  to  it  must  pass. 


CHAPTER  VI. 
EMPIRICAL  EQUATIONS. 

In  the  previous  chapters  I  have  discussed  some  of  the  methods  used  in 
plotting  curves  and  charts  from  given  equations.  The  present  one  will 
be  devoted  to  the  reverse  process,  namely,  the  derivation  of  equations  to 
fit  a  given  set  of  empirical  data  when  these  data  are  plotted  in  the  form  of 
a  curve  or  chart. 

The  subject  is  one  which  is  full  of  difficulties,  and,  so  far  as  I  know,  no 
systematic  general  method  has  ever  been  devised  which  will  give  the 
correct  form  of  equation  to  be  used.  The  discovery  of  the  equation's  form 
is  to  a  large  extent  a  matter  of  intuition  which  can  only  be  acquired  by 
long  experience.  Some  persons  seem  to  be  peculiarly  gifted  in  the  ability 
to  pick  out  the  proper  kind  of  equation  for  use  in  compensating  a  particular 
set  of  observations,  but  for  the  rank  and  file  of  the  men  engaged  on  experi- 
mental work  this  is,  and  probably  always  must  be,  a  matter  of  pure  guess- 
work, which  must  be  verified  by  cut-and-try  methods. 

In  getting  an  algebraic  expression  to  show  the  relations  between  the 
components  of  a  given  set  of  data  there  may  be  two  entirely  distinct  objects 
in  view,  one  being  to  determine  the  physical  law  controlling  the  results 
and  the  other  to  get  a  mathematical  expression,  which  may  or  may  not 
have  a  physical  basis,  but  which  will  enable  us  to  calculate  in  a  more  or 
less  accurate  manner  other  results  of  a  nature  similar  to  those  of  the 
observations. 

To  attain  the  first  result  it  will  generally  be  necessary  to  have  as  a 
starter  some  sort  of  hypothesis  as  to  the  physical  relations  of  the  data  in 
question,  although  in  a  few  isolated  cases  it  has  been  possible  to  arrive  at 
hitherto  unknown  laws  by  a  fortuitous  treatment  of  the  observations. 
In  such  a  case  as  this,  questions  as  to  the  intricacy  or  convenience  of  the 
formula  in  use  are  considered  subordinate  to  correctness  of  form. 

In  the  second  case,  where  we  want  an  expression  which  will  enable  us 
to  calculate  results  of  the  same  general  character  as  the  observations, 
form  will  generally  be  sacrificed  to  convenience  of  handling  and  no  pre- 
tense will  be  made  that  the  derived  formula  conforms  to  any  physical 
law.  This  condition  is  one  very  commonly  met  with  in  engineering 
practice,  and  will  be  the  one  with  which  this  chapter  is  chiefly  concerned, 

68 


EMPIRICAL   EQUATIONS  69 

It  has  been  a  common  matter  of  complaint  among  the  so-called  "  prac- 
tical" men  that  the  "theorists"  who  are  responsible  for  the  formulas  are 
very  prone  to  unnecessary  complication,  and  that  the  formulas  they  offer 
are  in  many  cases  no  more  exact  than  others  of  a  much  simpler  type. 
It  cannot  be  denied  that  there  is  some  justification  for  these  charges,  due, 
perhaps,  to  a  popular  impression  that  a  complicated  formula  presupposes 
brain  work  of  a  high  order  for  its  production. 

That  this  is  not  necessarily  true  needs  no  special  proof,  but,  on  the 
other  hand,  we  should  be  carefully  on  our  guard  lest  we  be  led  by  a  desire 
for  simplicity  into  devising  mere  rules  of  thumb,  applicable,  perhaps, 
to  the  very  special  conditions  in  which  they  originated,  but  nowhere  else. 
As  an  example  of  this,  take  the  numerous  formulas  which  have  been  pro- 
posed in  the  past  for  the  strength  of  gear  teeth;  formulas  giving  results 
which  in  some  instances  differ  from  each  other  by  several  hundred  per  cent. 

A  few  words  of  caution  may  be  necessary  at  the  start  to  prevent  the 
reader  from  expecting  too  much  of  the  processes  described.  Except  in 
some  of  the  simplest  cases  where  the  line  connecting  the  plotted  data  is 
straight,  it  will  generally  be  possible  to  fit  a  number  of  very  different 
forms  of  equation  to  the  same  curve,  none  of  them  exactly,  but  all  agree- 
ing with  the  original  about  equally  well.  Interpolation  on  any  of  these 
curves  will  usually  give  results  within  the  desired  degree  of  accuracy. 
The  greatest  caution,  however,  should  be  observed  in  exterpolation,  or 
the  use  of  the  equation  outside  of  the  limits  of  the  observations. 

If  the  form  of  the  equation  is  known  at  the  start  to  be  correct  and  the 
observations  are  merely  used  to  determine  the  constants,  exterpolation 
will  generally  be  safe.  If,  on  the  contrary,  the  form  of  the  equation  has 
been  guessed  at,  exterpolation  is  hazardous  in  the  extreme,  and,  if  an 
attempt  is  made  to  use  the  formula  much  outside  of  the  range  of  the 
observations  on  which  it  is  based,  serious  errors  may  be  committed. 

The  whole  subject  is  full  of  pitfalls  against  which  one  must  constantly 
be  on  guard. 

About  the  only  process  for  getting  empirical  equations  which  is  dis- 
cussed in  the  text-books  is  that  known  as  the  method  of  least  squares.  It 
will  yield  satisfactory  results  where  a  good  equation  has  been  chosen  at 
the  start,  but  it  is  tedious  and  laborious  in  the  extreme  even  under  the 
most  favorable  circumstances,  while  for  certain  forms  of  equation  the 
difficulties  of  the  method  are  so  great  that  it  can  hardly  be  considered  as 
practicable.  On  this  account,  and  because  it  can  be  found  fully  described 
in  the  ordinary  text-books,  I  shall  not  touch  upon  it  here,  but  confine  my- 
self to  a  number  of  graphical  or  semigraphical  methods  with  which  I  am 


70 


CONSTRUCTION  OF  GRAPHICAL  CHARTS 


acquainted.  Some  of  these  at  least  are  but  little  known.  Nevertheless, 
there  are  some  very  decided  advantages  in  their  use,  as  I  hope  to  show 
later. 


Square  Feet  of  Belt  Surface  per  Minute. 

.  i  s  §  §  i  i  i  i  i  i  1 

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FINDING  THE  EQUATION  FOR  A  STRAIGHT  LINE. 

To  begin  with,  let  us  take  a  very  simple  case  where  the  relation  between 
the  variables  in  the  equation  is  linear;  that  is,  where  the  plotted  results 
fall  upon  a  straight  line.  The  literature  of  engineering  contains  numerous 
examples  of  this  type,  and  I  have  chosen  as  illustrations  two  charts  taken 

from  Bulletin  No.  252  of  the 
University  of  Wisconsin,  entitled, 
"  Current  Practice  in  Steam 
Engine  Design,"  by  O.  N. 
Trooien.  Fig.  36  of  this  bulletin 
is  reproduced  here  in  Fig.  40, 
and  is  intended  to  show  the  rela- 
tion between  the  indicated  horse- 
power and  square  feet  of  belt 
surface  per  minute  for  Corliss 
engines. 

The  necessary  data  for  plot- 
ting this  diagram  were  obtained 
from  a  number  of  engine  builders. 
The  rim  speed  of  the  belt  pulley 

as  given  by  them  was  multiplied  by  the  width  of  the  belt  and  the  result  was 
used  as  an  ordinate,  while  the  horsepower  of  the  engine  was  taken  as  the 
abscissa.  A  point  was  thus  charted  for  each  engine,  as  shown  in  Fig.  40. 
In  this  case,  as  in  many  others  which  occur  in  practice,  the  chart  looks 
as  if  a  charge  of  bird  shot  had  been  fired  at  it,  and  it  is  manifestly  impossi- 
ble to  find  a  line  which  shall  even  approximately  pass  through  all  of  the 
points. 

If  we  have  any  reason  to  suppose  that  a  rational  formula  connecting 
the  belt  surface  and  horsepower  would  be  of  a  simple  linear  type,  all  we 
have  to  do  is  to  draw  a  straight  line  which  will  coincide  as  nearly  as  possible 
with  the  "axis"  of  the  group  of  points  and  take  its  equation  as  the  best 
representation  we  can  get  for  the  data.  In  the  case  of  horsepower  and 
belt  surface  it  is  generally  assumed  that  there  is  a  rough  proportionality 
between  them;  hence  a  straight  line  is  used  here.  The  points  in  Fig.  40 
show  two  fairly  distinct  groups  of  points,  and  judging  by  eye,  Mr.  Trooien 


100  200  300  400  500  COO  700  800  900  1000  1100 
Indicated  Horsepower 

FIG.  40. — Chart  showing  relation  between 
moving  belt  surface  and  horsepower  of  Corliss  en- 
gines. Equation  of  middle  line  is  y  =  2ix  + 1000. 


EMPIRICAL  EQUATIONS  7! 

appears  to  have  favored  the  lower  group  in  drawing  his  line  of  average 
values. 

This  proceeding  is  in  many  instances  not  only  justifiable,  but  impera- 
tive, if  we  wish  to  have  our  line  represent  the  best  probable  values.  It  often 
happens  that  certain  observations  are  known  to  be  more  accurately  made 
than  others,  and  hence  should  be  given  greater  weight  in  determining  the 
final  result.  In  the  least-square  method  the  better  observations  are 
affected  by  coefficients  corresponding  to  their  greater  accuracy  and  in 
the  graphical  method  the  same  end  is  attained  by  causing  our  line  to  pass 
closer  to  the  points  representing  the  better  observations.  Just  what  reason 
Mr.  Trooien  had  for  giving  greater  weight  to  one  group  than  to  the  other 
is  not  stated.  It  may  be  that  the  builders  had  a  better  reputation  or  the 
results  may  have  been  more  in  conformity  with  theoretical  considerations. 

Our  average  line  being  located  (and  it  will  generally  be  found  advan- 
tageous to  use  a  fine  thread  stretched  through  the  points  for  this  purpose) 
its  equation  is  easily  determined.  The  general  form  will,  of  course,  be, 

y  =  ax+b, 

where  b  is  the  height  of  the  intercept  on  the  Y-axis  (in  this  case  at  1000) 
and  a  is  the  tangent  of  the  angle  made  by  the  line  with  the  horizontal. 
The  Y-axis,  as  just  noted,  is  cut  at  1000,  and  the  ordinate,  through  1000 
horse  power  is  cut  at  22,000.  The  difference  is  21,000.  Dividing  this  by 
loco,  the  horizontal  distance,  gives  21  as  the  value  of  a. 

Our  formula  then  reads, 

y  =  2i  x+  1000, 

or,  as  y  represents  belt  speed  and  x  the  horsepower, 

5=2i  H. P.  +  1000. 

Where  the  points  representing  the  observations  scatter  as  badly  as  is 
the  case  here,  the  formula  must  be  looked  upon  as  a  very  rough  approxi- 
mation, and  considerable  deviation  from  it  may  be  allowed  in  practice 
when  for  any  reason  this  seems  desirable.  To  indicate  the  limits  within 
which  this  deviation  may  be  made  without  departing  from  common 
practice,  Mr.  Trooien  draws  two  lines  to  include  the  extreme  cases  and 
derives  the  constants  for  them  in  the  same  manner  as  before.  Since  all 
the  lines  meet  at  the  same  point  on  the  axis,  the  value  for  b  is  1000  in  each 
case,  while  a  varies  from  a  maximum  of  35  to  a  minimum  of  18.2. 

The  quantity  laid  out  on  the  X-axis  does  not  have  to  be  of  the  first 
power,  as  in  the  case  just  discussed,  and  may  even  be  itself  a  product  of 
several  variables.  In  such  a  case  we  must  lay  off  not  x  itself,  but  x*,  x3, 
*j  etc.,  as  the  case  may  be,  or  x  z  if  it  is  a  product. 


72 


CONSTRUCTION   OF   GRAPHICAL   CHARTS 


ANOTHER  ILLUSTRATION  OF  FINDING  THE  EQUATION  FOR  A  STRAIGHT  LINE. 

This  may  be  illustrated  by  the  chart  for  the  connecting  rods  of  Corliss 
engines  shown  in  Fig.  16  of  the  same  bulletin  and  reproduced  here  in 
Fig.  41. 

If  the  Euler  formula  for  struts  be  taken  as  correct  for  the  connecting 

rod,  it  may  be  reduced  to  the 
expression. 

d=C\/DL 

where  d  is  the  diameter  of  the 
rod,  C  a  constant  whose  value 
is  to  be  determined,  D  the 
diameter  of  the  piston  (supposed 
to  be  acted  upon  by  a  standard 
steam  pressure),  andZ,  the  length 
of  the  rod. 

From  the  data  furnished  by 
the  engine  builders  the  points  in 
the  chart  shown  in  Fig.  41  were 
plotted,  using  the  values  of  d  as 
ordinates  and  of  \/DL  as  abscissas.  The  resulting  line  should  be 
straight  and  pass  through  the  origin,  and  the  angle  with  the  horizontal 
gives  the  desired  value  of  Cas  5.5/60,  or  0.092,  for  the  mean  and  0.104 
and  0.081  as  the  maximum  and  minimum  values. 


30  40 

Values  of  V^DL 


50 


FIG.  41. — Chart  showing  relation  between 
diameter  of  connecting  rod  and  square  root 
of  piston  diameter  times  the  length  of  rod  for 
Corliss  engines.  Equation  of  middle  line  is 


FINDING  THE  EQUATION  OF  A  CURVE. 

Next  let  us  consider  the  case  where  the  line  connecting  the  observations 
is  curved.  Here  we  have  no  ready-made  equation  as  with  the  straight 
line,  requiring  merely  the  discovery  of  a  couple  of  constants.  The  general 
form  of  the  equation  must  be  guessed  at  if  the  physical  law  is  unknown, 
and  here  we  encounter  one  of  the  greatest  difficulties  connected  with  the 
subject  and  one  for  which  it  is  practically  impossible  to  offer  much  real  help. 

The  appearance  of  the  curve  may  or  may  not  afford  a  clue,  and  in  this 
connection  it  is  suggested  that  a  book  like  Frost's  "Curve  Tracing," 
may  be  useful  for  reference;  It  contains  a  large  number  of  curves  plotted 
from  various  equations  and  their  shapes  will  sometimes  suggest  a  good 
form  of  equation  if  we  are  at  fault. 

It  has  sometimes  been  suggested  as  a  solution  of  this  difficulty  that  we 
plot  a  considerable  number  of  functions,  such  y=x*,  y=xs,  y  =  log.  x, 
y=i/x,  etc.,  on  a  straight  line  and  then  from  any  pole  draw  a  series  of 


EMPIRICAL   EQUATIONS  73 

radiating  lines  through  the  points  thus  found  as  shown  in  Fig.  42  where 
y=oc2  has  been  used.  The  observed  results  are  plotted  on  a  similar 
straight  line  for  equally  spaced  values  of  the  variable.  This  graduated 
line  is  then  laid  on  the  radiating  lines  and  shifted  around  until  we  get 
the  plotted  points  falling  on  them.  Such  an  agreement  would  indicate 
at  once  the  proper  function  to  use,  V  v  \  i  /  / 

^^iVQ       \3  \4  5/ 6^ 

and  a  measurement  of  its  distance 
from  the  pole  would  indicate  the 
coefficient. 

While  this  looks  promising,  my 

Own  experience  leads  me  to  accord     FlG-  42.— Trial  diagram  of  a  known  function. 

In  this  case  y  =x2. 

it  but  little  practical  value.     The 

observation  points  can  hardly  ever  be  made  to  agree  even  approximately 

with  the  trial  function. 

Many  experimenters  assume  that  an  equation  of  the  parabolic  form, 

y  =  a+b oc+c oc2.  .  .  .  etc. 

may  be  used  for  almost  any  class  of  observations  with  good  results,  and 
it  is  surprising  sometimes  how  closely  it  may  be  made  to  fit  unpromising 
conditions. 

It  should  not,  however,  be  blindly  used  for  all  cases,  for  while,  on  the 
one  hand,  it  may  be  forced,  with  a  sufficient  number  of  terms,  into  the 
semblance  of  an  agreement  with  almost  any  set  of  data,  on  the  other  hand, 
a  large  number  of  terms  is  detrimental  to  its  subsequent  use  in  calculation 
and  in  many  cases  a  far  simpler  equation  may  be  discovered  which  will 
not  only  be  easier  to  handle,  but  may  even  give  more  accurate  results. 
For  instance,  the  crest  of  a  sine  curve  may  be  made  to  agree  quite  closely 
with  a  parabola,  but  the  longer  this  arc  is  the  greater  is  our  difficulty  in 
getting  a  fit. 

In  fitting  an  equation  to  a  given  set  of  observations  the  first  step  is  to 
draw  through  the  plotted  points  a  smooth  curve.  If  the  experimental 
work  has  been  carefully  and  accurately  done  the  curve  may  be  made  to 
pass  through,  or  close  to,  almost  all  the  points.  If  not,  the  curve  must  be 
drawn  in  such  a  way  as  to  represent  a  good  probable  average ;  that  is,  so 
as  to  leave  about  an  equal  number  of  points  at  about  equal  distances  on 
either  side  of  it,  these  distances,  of  course,  being  kept  as  small  as  possible. 
Such  a  curve  is  assumed  to  represent  the  most  probable  values  of  the 
observations,  and  we  then  attempt  to  get  its  equation. 

It  may  be  stated  that  it  is  always  possible  to  get  an  equation  which 
will  agree  exactly  with  a  given  curve  at  any  desired  number  of  points, 
providing  we  use  an  equal  number  of  constants  in  our  equation. 


74 


CONSTRUCTION  OF  GRAPHICAL  CHARTS 


17000 

16000 
15000 
14000 
13000 
12000 
11000 
10000 

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7000 

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6000 
-4000 
3000 
2000 
1000 

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1                8                0               10              11              12               13              14              15              16              17               18              M 

Speed  in  Knots 


FIG.  43. --Chart  showing  relation  between  indicated  horsepower  and  speed  in  knots  for  the 
battleship  "Maine."     Equation  of  the  dashed  curve  is  y  =  440.53;  —  82. 


EMPIRICAL   EQUATIONS  75 

METHOD  OF  SELECTED  POINTS. 

This  is  called  the  method  of  selected  points  and  will  be  described 
first  as  it  is  the  simplest  and  quickest  method  and,  if  a  good  equation 
has  been  chosen  at  the  start,  we  may  get  results  of  a  very  satisfactory 
character. 

For  purposes  of  illustration  I  have  chosen  a  curve  given  in  the 
Journal  of  American  Society  of  Naval  Engineers  for  November,  1902. 
It  shows  the  relation  between  the  speed  in  knots  and  the  indicated  horse- 
power for  the  battleship  "  Maine"  and  is  reproduced  by  the  solid  line 
in  Fig.  43. 

The  data  from  which  the  curve  was  plotted  are  not  given  and  there  is 
no  means  of  knowing  how  accurately  it  represents  the  results  of  the  test. 
It  will,  therefore,  be  taken  as  it  stands  and  an  attempt  made  to  find  the 
compensating  equation.  As  to  the  form  of  the  equation,  we  will  disre- 
gard all  theoretical  considerations  and  assume  it  to  be  parabolic  since 
it  has  most  of  the  ear-marks  of  this  type. 

The  curve  stops  at  about  eight  knots  and  we  have  nothing  to  guide 
us  as  to  its  shape  below  this  point.  The  assumption  will  be  made, 
however,  that  the  horsepower  and  speed  became  zero  together;  that 
is,  that  the  curve  passes  through  the  origin.  If  this  is  so  the  first 
constant  in  the  general  parabolic  equation  (the  one  unattached  to  a 
variable)  vanishes. 

Let  us  assume  that  the  equation  contains  the  first  three  powers  of  #, 
or  that 


where  y  represents  the  indicated  horsepower  and  oc  the  speed  in  knots. 
We  have  here  three  constants  whose  values  must  be  determined.  To 
do  this  take  three  points  on  the  curve,  one  at  about  the  middle  and  the 
others  at  or  near  the  ends,  and  form  three  equations,  inserting  in  them 
the  values  of  y  and  x  for  these  points  taken  from  the  curve. 

In  the  case  in  question  I  have  selected  the  points  at  9,  13,  and  17 
knots.  The  corresponding  values  for  y  (the  horsepower)  are  1400,  4180, 
and  11,350. 

Inserting  these  in  the  chosen  equation  we  have: 

1,400=   9  a+   81  b+    729  c, 

4,180=13  0+169  £+2,197  c, 

11,350=17  0+289  &  +  4,9I3  c- 

These  equations  are  solved  by  the  customary  methods  for  a,  b,  and  c, 
giving  us  440.5  for  a,  —82.32  for  &,  and  5.628  for  c. 


76  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

The  equation  then  reads: 

^  =  440.5  ^-82.32  x2+  5.628  *3, 
or 

H.  P.  =  440.5  5-82.32  S2+  5.628  S3. 

The  curve  for  this  equation  has  been  drawn  as  a  broken  line  on  the 
same  chart  as  the  original  curve,  and  is  seen  to  pass  through  the  chosen 
points  exactly  and  to  give  a  very  fair  agreement  at  nearly  every  other 
point. 

At  the  upper  end,  however,  although  the  two  curves  are  not  much 
separated,  there  is  a  considerable  difference  in  the  horsepower  as  read 
from  the  two  curves,  and  the  indications  are  that  this  will  become  worse 
as  we  overstep  the  limits  of  the  chart.  Up  to  about  171/2  knots,  however, 
the  equation  would  usually  be  considered  a  passable  fit.  The  rapid  rise 
in  the  horsepower  as  the  speed  increases  at  the  upper  end  of  the  curve 
would  indicate  that  better  results  might  have  been  reached  by  the  use  of 
a  higher  power  of  x  in  the  equation. 

ANOTHER   ILLUSTRATION  OF  THE  METHOD  OF  SELECTED  POINTS. 

The  above  method  will  answer  every  requirement  in  many  cases,  but 
too  much  reliance  should  not  be  placed  in  it  without  an  actual  test  of  the 
results.  As  an  example  of  the  danger  of  this  I  have  applied  the  method 
to  a  series  of  experiments  showing  the  variation  of  the  coefficient  of  fric- 
tion of  straw-fiber  friction  drives  with  the  slip. 

The  experiments  were  made  by  Professor  Goss,  who  describes  them 
in  the  Transactions  of  the  American  Society  of  Mechanical  Engineers 
for  1907,  page  1099. 

To  avoid  a  confusion  of  notation,  I  have  replotted  the  curve  from  the 
original  paper  in  Fig.  44,  with  the  ordinates  and  abscissas  interchanged. 
The  small  circles  represent  the  observations  and  the  solid  curve  is  Pro- 
fessor Goss'  idea  of  the  best  representation  of  their  average  value.  We 
will  attempt  to  compensate  this  curve  by  a  suitable  equation. 

.  At  first  glance  the  curve  seems  to  have  some  of  the  characteristics 
of  the  parabolic  type,  enough  at  any  rate  to  make  it  amenable  to  treatment 
by  that  form  of  equation.  It  straightens  out  suspiciously,  however,  in 
each  direction,  as  it  leaves  the  region  of  greatest  curvature  near  the  ordi- 
nate  erected  at  0.4,  and  this  would  suggest  the  hyperbolic  rather  than  the 
parabolic  type.  As  an  experiment,  however,  we  will  run  it  out  on  the 
assumption  of  its  being  a  parabola  and  will  try  compensating  by  the 
equation 


EMPIRICAL   EQUATIONS 


77 


The  four  constants  will  make  it  possible  to  get  four  points  of  exact 
agreement  instead  of  three,  as  in  the  previous  example,  and  we  should 
naturally  expect  that  the  general  agreement  would  be  better  on  account 
of  this  larger  number  of  points. 

Let  these  points  be  7  =  0.55,  #  =  0.15;  7  =  0.825,  #  =  0.3;  7=1.42, 
#  =  0.4;  7  =  2.7,  #  =  0.45.  The  four  equations  then  become: 

0.55   =  0  +  0.15  6  +  0.0225  £  +  0.003375  d,  ' 

0.825  =  0  +  0.3  6  +  0.09  £  +  0.027  d,  /j 

1.42  =  0  +  0.4  6  + 0.16  £  +  0.064  d,  j\ 

2.7     =0  +  0.45  6  +  0.2025  £+0.091125  d.  /' 

/  i 


0.2  0.3 

Coefficient  of  Frictiou 


FIG.  44.— Chart  showing  relation  between  coefficient  of  friction  and  slip  for  straw-fiber 
frictions.     Equationisy=          '        +0.181. 

The    solution   of    these  equations   gives   us   0  =  —5.89,   b  =  80.5, 
c  =  -  308,  d  =381.8, 
or, 

y=   —5.89  +  80.5^  —  308  X2  +  381.8  x*. 

The  values  of  7  were  now  calculated  for  every  0.05  of  x  from  o.i  to 
0.5,  and  the  result  is  shown  by  the  broken  line  on  the.  same  diagram.     It 


78  CONSTRUCTION    OF    GRAPHICAL   CHARTS 

hits  the  selected  points  with  practical  exactness,  but  it  would  require 
a  vivid  imagination  to  say  that  the  fit  elsewhere  was  even  fairly  good.  A 
larger  number  of  selected  points  and  constants  would  undoubtedly  have 
helped  materially  in  improving  this  state  of  affairs,  but  the  most  cursory 
inspection  of  the  diagram  will  show  that  the  trouble  is  not  due  to  the 
small  number  of  points,  but  rather  to  the  choice  of  an  improper  form 
of  equation. 

Returning  now  to  the  suggestion  made  above  as  to  its  hyperbolic  form, 
let  us  see  what  can  be  done  on  that  supposition.  We  will  assume  that  the 
curve  is  a  rectangular  hyperbola  of  which  we  do  not  know  the  asymptotes. 

Let  us  try  an  equation  of  the  form 

(y+a)(x+b)-*c. 

The  three  constants  will  demand  three  equations,  and  we  will  select 
for  our  points  ^  =  0.55,  #  =  0.15;  ^  =  0.825,  #=0.3;  ^  =  2.7,  #  =  0.45. 

Substituting  these  values  in  the  equation  above  we  have, 


(0.825  +  a)  (0.3  +  &)  =c, 
(2.7  +  a)  (0.45  +  b)  =c. 

The  solution  of  these  equations  for  a,  b,  and  c,  gives  us 
a=  —0.1806,6=  —0.5015,  and  c=  —0.1298. 

These  values,  in  round  numbers,  substituted  in  the  original  equation 
give  us 

(y-o.i8i)  (^-0.502)  =  -0.13, 
or, 

—  0.11 

y  =  -  —  +0.181. 
x  —  0.502 

If,  now,  we  substitute  values  of  oc  for  every  0.05  from  0.15  to  0.45, 
we  get  the  points  represented  by  the  double  circles  in  the  chart.  They 
agree  so  closely  with  the  original  curve  as  to  be  practically  identical  with 
it.  Thus,  with  a  less  number  of  points  we  have  obtained  an  extremely 
satisfactory  fit,  and  have  given  a  practical  illustration  of  the  statement 
made  above  as  to  the  desirability  of  starting  with  a  good  equation  rather 
than  trying  to  force  a  fit  by  the  use  of  an  unsuitable  equation  and  a  large 
number  of  constants. 

VALUE  OF  LOGARITHMIC  CROSS-SECTION  PAPER  IN  DETERMINING  FORM 
AND  CONSTANTS  OF  AN  EQUATION. 

This  may  be  a  good  place  to  say  that  the  logarithmic  paper  described 
in  a  previous  chapter  is  often  of  great  service  in  determining  the  form  and 
constants  of  an  equation. 


EMPIRICAL   EQUATIONS  79 

If  the  equation  involves  only  a  simple  product  or  quotient  with  no 
addition  or  subtraction,  its  trace  on  logarithmic  paper  will  be  a  straight 
line.  The  tangent  of  the  angle  made  by  this  line  with  the  horizontal 
(and  this  may  be  positive  or  negative)  will  give  the  exponent  of  the  vari- 
able, while  the  intercept  on  the  Y-axis  will  give  the  constant  by  which  the 
variable  is  multiplied. 

It  is  much  to  be  regretted  that  the  ordinary  commercial  logarithmic 
paper  is  only  laid  off  from  i  to  10  on  the  axes,  for  my  experience  is  that 
almost  invariably  the  line  will  extend  beyond  these  limits,  and  it  then 
becomes  difficult  to  see  clearly  if  it  is  rectilinear,  since  it  must  be  broken 
and  appear  in  two  or  more  places  on  the  sheet.  If  such  paper  were 
printed  with  graduations  on  each  axis  from  i  to  100  instead  of  from  i  to 
10,  it  would  greatly  facilitate  many  of  these  operations.  Any  curve  having 
the  aspects  of  the  hyperbolic  or  parabolic  type  should  always  be  so  plotted, 
since,  if  it  does  appear  as  a  straight  line,  it  saves  a  large  amount  of  labor 
in  determining  its  equation. 

One  special  case  may  be  mentioned  here,  which  is  sometimes  useful 
in  gas-engine  work;  namely,  the  determination  of  the  exponent  of  the 
v  in  the  equation  for  the  expansion  curve.  If  we  have  an  indicator 
diagram  we  take  the  ordinates  representing  the  pressures  (absolute)  and 
lay  them  out  on  the  logarithmic  paper  from  points  on  the  X-axis  repre- 
senting the  volumes  (which  must  include  the  clearance).  The  points 
thus  found  should  fall  upon  a  line  which  is  sensibly  straight  if  the  exponent 
is  constant  for  all  parts  of  the  curve.  Otherwise  the  exponent  must  be 
determined  for  any  particular  point  by  drawing  the  tangent  to  the  curve 
there. 

As  an  illustration,  I  have  reproduced  the  expansion  line  from  the  indi- 
cator diagram  of  an  old  Clerk  gas  engine.  The  volumes  are  measured 
from  the  clearance  line  in  any  convenient  unit.  The  length  of  the  diagram 
made  it  convenient  to  call  the  clearance  volume  9.  From  there  on,  the 
indicator  diagram  was  divided,  as  shown  in  Fig.  45  (a),  and  the  logarithms 
corresponding  to  the  numbers  on  the  X-axis  were  laid  off  on  the  X-axis  of 
the  lower  logarithmic  diagram,  (b)  of  Fig.  45. 

The  pressures  from  the  absolute  zero  were  then  measured  from  the 
indicator  card  and  their  logarithms  laid  off  from  the  corresponding  points 
of  the  X-axis  of  (b). 

A  straight  line  was  now  drawn  to  indicate  the  general  direction  of  the 
middle  set  of  points  and  then  a  parallel  to  it  through  10  on  the  X-axis. 
Its  intercept  on  the  Y-axis  measured  in  linear  (not  logarithmic)  units 
gives  the  tangent  of  the  angle  of  slope.  In  this  case  it  is  1.32  which, 


8o 


CONSTRUCTION  OF  GRAPHICAL  CHARTS 


divided  by  i  (the  distance  to  10  measured  on  X),  gives  1.32  as  the  value 
of  the  exponent. 

The  method  of  selected  points,  while  accurate  enough  for  many  pur- 
poses, especially  where  the  form  of  the  equation  is  definitely  known  at  the 


10    12     14     16    18    20    22    24    26 


0  10    12   14   16  18  20  22  24  25 

FIG.  45. — Expansion  line  of  a  gas-engine  indicator  card  and  logarithmic  determination  of 
value  of  exponents  in  the  equation  of  the  expansion  curve. 

start,  is  not  so  satisfactory  when  we  wish  for  greater  refinement,  and 
especially  when  we  are  in  the  dark  as  to  the  proper  form  of  equation.  The 
number  of  points  which  can  influence  the  result  is  no  more  than  the 
number  of  constants  employed,  and  if  we  wish  to  use  a  small  number  of 


EMPIRICAL   EQUATIONS  8 1 

constants  we  cannot  expect  any  high  degree  of  accuracy  in  the  fit.  Some 
method  by  which  a  larger  number  of  points  on  our  curve  may  enter  into 
the  result  without  burdening  the  equation  with  constants  is,  therefore, 
much  to  be  desired. 

METHOD  OF   EQUATING  THE  AREA  AND  MOMENTS   OBTAINED  FROM 

MEASURING  THE  AREA  UNDER  A  CURVE  WITH  THE 

INTEGRATION  OF  THE  ASSUMED  EQUATION 

OF  THE  CURVE. 

Suppose  that,  an  observation  curve  being  drawn,  we  obtain  its  area  by 
any  planimetric  method.  If,  now,  we  find  the  area  of  the  curve  of  the 
assumed  equation  by  integration  and  equate  it  to  the  area  just  found  of 
the  observation  curve,  we  evidently  have  a  condition  in  which  we  can  take 
account  of  as  large  a  number  of  points  as  we  please  without  necessarily 
using  a  large  number  of  constants.  In  fact,  this  one  equation  takes  care 
of  one  and  only  one  constant.  It  would,  of  course,  be  possible  to  have 
two  curves  of  equal  area  and  quite  different  shape  if  the  assumed  formula 
were  not  well  chosen. 

Suppose,  however,  that  we  get  the  moment  of  the  area  of  the  original 
curve  by  dividing  it  up  into  a  number  of  vertical  slices,  taking  the  area  of 
each  slice  above  the  X-axis  and  multiplying  it  by  the  distance  of  its  center 
from  any  arbitrary  vertical  axis,  generally  Y,  and  then  adding  the  moments 
thus  found;  we  shall  in  this  way  obtain  the  moment  about  the  assumed 
axis  of  the  entire  area  between  the  curve  and  X.  Its  value  will  evidently 
depend  upon  the  form  as  well  as  the  area  of  the  curve.  The  moment  of 
the  assumed  curve  may  likewise  be  determined  by  integration  and  can  be 
placed  equal  to  the  measured  moment.  This  accounts  for  another  con- 
stant. Similarly  we  may  obtain  second  and  third  moments,  etc.,  by 
multiplying  the  areas  of  the  slices  by  the  square  and  cube  of  the  distances 
from  the  assumed  axis  and,  from  each  of  these,  form  equations  with  the 
same  moments  of  the  theoretical  curve.  We  must,  of  course,  have  as 
many  of  these  equations  as  we  have  constants  to  determine. 

Any  of  the  well-known  methods  for  getting  the  areas  and  moments 
may  be  used,  but  as  it  will  make  the  explanation  simpler  I  shall  get  my 
areas  and  moments  in  what  follows  by  taking  the  mean  ordinates  of  a 
series  of  vertical  slices  in  the  same  way  that  we  do  when  averaging  an 
indicator  diagram,  and  assume  that  all  necessary  accuracy  can  be  secured 
by  making  the  strips  narrow  and  of  considerable  number.  As  an  illus- 
tration of  the  application  of  the  method  it  will  be  interesting  for  purposes 


82 


CONSTRUCTION  OF  GRAPHICAL  CHARTS 


of  comparison  to  take  again  the  curve  for  the  speed  and  horsepower  of 
the  "  Maine." 

The  same  formula  will  be  assumed  as  before,  having  three  constants 
to  be  determined  and,  therefore,  demanding  three  equations.  The  curve 
extends  practically  from  8  to  18  knots  and  these  will  be  taken  as  the 
limits  within  which  to  work. 

For  convenience  we  will  divide  this  space  into  10  vertical  slices.  A 
greater  number  would  lead  to  greater  accuracy,  but  the  work  of  calcula- 
tion is  laborious  at  the  best  and  for  illustrative  purposes  this  will  be 
amply  sufficient.  The  height  of  the  curve  at  the  middle  of  each  of  these 
spaces  is  then  measured  and  tabulated  in  the  column  headed  y  alongside  of 
the  corresponding  value  of  x.  Next  to  the  x  column  is  one  of  x2.  Then 
follow  columns  for  oc  y  (the  first  moment)  and  x2  y  (the  second  moment) . 


y 

x 

x2 

xy 

x2y 

1,260 

8-5 

72.25 

10,710 

91,077 

1,660 

9-5 

90.25 

J5>77o 

149,810 

2,150 

10.5 

110.25 

22,575 

237,040 

2,850 

ii.  5 

132.25 

32,775 

376,920 

3,670 

12.5 

156.25 

45,875 

573,440 

4,730 

13-5 

182.25 

63,855      • 

862,040 

6,050 

14-5 

2IO.  25 

87,725 

1,272,000 

7,750 

15-5 

240.25 

120,130 

1,862,000 

10,000 

16.5 

272.25 

165,000 

2,722,500 

13,050 

17-5 

306.25 

228,370 

3,996,5oo 

53,170  =  Area 

792,785  =  Mj. 

12,143,327  =  M2 

Since  the  width  of  the  slices  is  i,  the  height  of  the  middle  ordinate 
gives  us  its  area  at  once,  and  the  sum  of  the  ordinates  will  be  the  area  of 
the  space  under  the  entire  curve. 

The  sum  of  the  values  in  the  y  column  gives  us  53,170  as  the  area. 
The  sum  of  the  values  of  x  y  gives  us  792,785  as  the  first  moment,  and 
the  sum  of  the  values  of  oc2  y  gives  us  12,143,327  as  the  second  moment. 
Both  moments  are  reckoned  about  the  Y-axis. 

These  quantities  must  be  equated  to  the  area  and  first  and  second 
moments  of  the  assumed  theoretical  curve,  to  get  which  we  must  use  a 
little  integral  calculus.  The  area  of  a  small  vertical  slice  of  height  y 
and  width  d  x  is  y  d  oc  and  since, 

y  =  a  x+b  x2  +  c  x3, 
y  dx  =  ax  dx+bx2  dx  +  cx*  dx. 


EMPIRICAL   EQUATIONS  83 

The  integral  of  this  expression  is  the  area  of  the  curve,  or 

/x 
2  a  x  d  x  +  b  oc2  d  x  +  c  x\  d  x} 
*i 
where  x1  and  x2  are  the  limits  between  which  the  integration  is  to  be 

performed  (here  8  and  18),  or 

A  =  -(*i-*^  +  *(*|-  *?)  +  -(*£  -*?)• 
234 

This,  after  substituting  the  values  of  xl  and  x2  given  above,  is  placed 
equal  to  53,170. 

If  we  multiply  the  differential  area  y  d  x  by  x  we  get  its  moment  about 
the  Y-axis  and  its  integral  will  be  the  first  moment  of  the  entire  area,  or 


=  £°C 
J 


45 
This  is  placed  equal  to  the  measured  first  moment,  or  792,785. 

Multiplying  the  differential  area  next  by  x2,  we  get  its  second  moment 
about  the  Y-axis  and  its  integral  will  be  the  second  moment  of  the  whole 
area,  or 

J/v*  fl  i)  C 

"  2ax3dx+bx*dx+cx5dx=-(x%—x$)  +  -(x%—xl)  +  ~(x$—xf)y 
xl  450 

which  must  be  placed  equal  to  12,143,327. 

After  substituting  the  limiting  values  of  x±  and  x2,  which  are  8  and  18, 
we  have  the  three  equations 

1300+1773  6+25,  218^  =  53,170, 

1773  0+25,218  &+37i,366c  =  792,  785, 

25,218  0+371,366  b+  5,624,977  c=  12,143,327. 

In  solving  these  equations,  while  it  may  not  be  necessary  to  run  all 
calculations  out  to  the  last  figures,  it  will  generally  be  desirable  to  carry 
them  out  to  five  or  six  significant  figures,  since  we  often  have  to  take  the 
difference  between  two  numbers  of  nearly  equal  magnitude,  in  which 
case  the  last  figures  may  have  an-  important  influence  on  the  result. 
The  slide  rule  is,  therefore,  absolutely  useless  for  these  calculations 
except  as  a  check  against  large  errors.  After  the  calculations  are  complete 
it  will  generally  be  safe  to  throw  away  all  except  the  first  three  or  four 
significant  figures  in  order  to  simplify  the  formula  for  practical  use. 
The  solution  of  the  above  equations  gives 

0  =  422.8,  £=  -77.98,  ^=5.4115, 
making  the  equation  read 

^  =  422.8  #-77.98  x*+  5.4115  **> 
or,  more  simply, 

^  =  423  #-78  x2+  5.41  xs. 


84  CONSTRUCTION    OF   GRAPHICAL   CHARTS 

By  the  method  of  selected  points  we  got 

^  =  481^-88.5^+5.853^. 

The  agreement  is  as  close  as  could  be  expected  and  is  really  closer 
than  the  appearance  of  the  equation  might  lead  us  to  suppose. 

This  is  shown  in  Fig.  43,  where  the  equation  just  obtained  is 
plotted  with  the  previous  one  by  selected  points.  As  the  curves  run 
pretty  close  together,  I  have  not  attempted  to  draw  the  last  one,  but 
have  simply  indicated  the  value  of  y  for  each  even  knot  by  a  small 
circle. 

The  difference  between  the  two  curves  is  quite  small,  the  last  one  being 
possibly  slightly  nearer  the  curve  we  are  trying  to  compensate  than  the  first. 
So  small  a  difference  would  hardly  make  it  worth  while,  as  a  rule,  to  use 
the  more  laborious  method  of  moments  if  we  knew  that  the  results  were 
going  to  come  out  this  way  beforehand.  We  have  no  means  of  knowing 
this,  however,  and  there  is  generally  an  added  feeling  of  safety  in  using  it 
on  account  of  the  larger  number  of  points  which  are  taken  account  of. 
We  should  probably  have  obtained  a  closer  approximation  to  the 
original  curve  by  using  a  larger  number  of  ordinates  in  getting  our 
area  and  moments.  Whether  or  not  this  would  be  desirable  would 
have  to  be  determined  after  an  inspection  of  the  calculated  curve  to 
see  if  its  deviation  from  the  original  was  within  the  desired  limits  of 
accuracy. 

This  method  is  of  very  general  application  and  may  be  used  for  any 
equation  of  integrable  form. 

AN  ALINEMENT  CHART  METHOD. 

The  next  method  I  propose  to  discuss  is  one  based  on  the  alinement 
chart  described  in  Chapter  II. 

The  method  is  due  to  Captain  Batailler,  of  the  French  artillery  service, 
who  describes  the  process  in  the  Revue.  d'Artillerie  of  December,  1906. 
Those  who  are  interested  are  referred  to  it  if  they  desire  fuller  information 
than  can  be  given  in  this  brief  outline.  The  process  depends  on  the 
alinement  of  a  series  of  points  taken  from  the  data  or  from  a  curve  which 
is  assumed  to  represent  them. 

It  is  not  easy  to  explain  the  method  in  a  simple  manner,  but  I  hope 
that  I  shall  at  least  be  able  to  make  the  practical  application  clear.  This 
I  think  can  be  best  done  by  working  out  a  practical  example,  explaining 
each  step  as  it  is  taken. 

The  example  chosen  will  be  the  data  given  by  Prof.  R.  T.  Stewart  as 


EMPIRICAL   EQUATIONS  85 

the  results  of  his  experiments  on  the  collapsing  pressure  of  bessemer- 
steel  tubes,  and  published  in  the  Transactions,  of  the  American  Society  of 
Mechanical  Engineers  for  1906.  Professor  Stewart  showed  his  results  in 
chart  form  by  laying  off  the  values  of  the  thickness  of  the  tube  divided  by 
the  diameter,  or  tjd,  on  the  X-axis  and  the  corresponding  collapsing  pres- 
sures as  ordinates.  He  found  that  a  smooth  curve  drawn  through  these 
points  was  difficult  to  represent  by  any  simple  formula  and,  therefore, 
took  two  bites  at  it,  so  to  speak,  and  derived  two  formulas  limited  in  their 
application  to  different  parts  of  the  field.  This  is  a  very  common  and 
useful  expedient  where  the  experimental  curve  is  rebellious  to  representa- 
tion by  a  simple  formula. 

Let  us  see  what  can  be  done  toward  getting  the  whole  range  of 
results  into  one  equation.  To  start  with,  the  averages  from  the 
tabulated  results  have  been  plotted  in  Fig.  46,  and  are  indicated  by 
the  small  circles.  In  doing  this  I  have  interchanged  the  ordinates  and 
abscissas  as  they  appear  in  Professor  Stewart's  chart  since,  with  the 
form  of  equation  I  wish  to  try,  there  might  otherwise  be  some  confusion 
of  nomenclature. 

Then  a  smooth  curve  was  passed  through  these  points  so  as  to 
represent  as  nearly  as  possible  a  good  general  average.  This  is  not 
strictly  necessary  for  the  first  process  I  am  going  to  describe,  but  I 
have  done  it  in  order  to  have  something  definite  to  work  toward  as  a 
measure  of  the  success  of  the  method,  and  also  because  a  definite 
curve  is  more  suggestive  of  the  type  of  equation  than  a  number  of  scat- 
tered points.  Professor  Stewart  assumed  that  the  greater  part  of  the 
curve  is  a  straight  line  with  a  sharp  bend  toward  the  origin  as  the  lower 
values  are  approached. 

He  was  probably  justified  in  doing  this,  as  the  small  number  of  obser- 
vations among  the  higher  values  make  the  direction  of  the  curve  in  that 
region  somewhat  uncertain. 

In  my  chart  I  have  drawn  the  line  with  a  reversal  of  curvature  to  per- 
mit it  to  pass  closer  to  the  higher- value  observations  and  thus  get  a  some- 
what closer  agreement  with  the  actual  tests.  At  the  lower  end  of  the 
curve,  according  to  Professor  Stewart,  the  collapsing  pressure  seems  to 
vary  as  (//d)3,  or  t/d  is  proportional  to  the  cube  root  of  the  collapsing 
pressure.  Acting  on  this  hint,  we  will  use  <\/#  in  our  equation  (x  being 
taken  to  represent  the  collapsing  pressure) . 

Now,  the  curve  as  I  have  drawn  it  reverses  its  direction  of  curvature 
as  it  moves  away  from  the  origin.  This  effect  could  be  brought  about 
by  the  use  of  some  power  of  x  in  the  equation  in  addition  to  the  root. 


86 

0.09 
0.06 
0.07 
0.06 

0.05 

V|T. 

•5 

!..« 

0.03 
0.08 

0.01 

• 

CONSTRUCTION   OF   GRAPHICAL  CHARTS 

i 

^ 

P 

/ 

i 

/ 

/     , 

/ 

/ 

) 

/ 

/ 

A 

5 

./ 

/ 

/ 

f   O 

0 

/ 

? 

1 
/ 

I 

/ 

0 

1000 


5000 


2000  3000  4000 

Collapsing  Pressures 

FIG.  46. — Chart  showing  relations  between  collapsing  pressure  of  bessemer  steel  tubes  and  the 
ratio  of  thickness  to  diameter.    Equation  of  curve  is  —  =  o. 002 74  *?/ P  +0.000000001  iP2. 


EMPIRICAL  EQUATIONS  87 

The  power  will  have  but  little  influence  on  the  shape  of  the  curve  for 
the  lower  values  of  x  where  the  predominant  effect  of  the  root  is  felt,  but 
as  we  get  to  the  higher  values  the  power  will  overbalance  the  effect  of  the 
root  and  cause  the  reversal  we  wish.  A  high  power  is  evidently  not  indi- 
cated as  the  bend  upward  is  comparatively  small,  hence  (as  it  is  easily 
calculated)  we  will  try  the  second. 

Let  our  trial  equation  then  take  the  form 

y  =  A  ^/x+Bx2 
where  y  represents  t/d  and  x  the  collapsing  pressure. 

For  convenience  in  handling  let  us  express  these  pressures  in  units  of 
1000  pounds. 

The  general  form  of  equation  used  in  the  discussion  of  the  alinement 
diagram  was 

au+b  v  =  c, 

where  u  and  v  represent  measured  distances  on  the  U-  and  V-axes.  If 
u  and  v  are  kept  constant  while  a,  6,  and  c  vary,  we  get  a  series  of  points 
lying  along  the  straight  line  joining  u  and  v.  Hence,  if  this  line  can  be 
determined,  its  intersection  with  the  U-  and  V-axes  should  fix  the  values 
of  u  and  v. 

In  our  assumed  formula  A  and  B  are  constants,  therefore  let  us 
consider  that  they  replace  the  quantities  u  and  v  in  the  alinement 
equation.  Now  <f/x,  x2,  and  y  may  be  given  various  values,  hence 
let  us  suppose  that  they  take  the  place  of  a,  b,  and  c  in  the  alinement 
equation. 

In  order  to  get  the  position  of  the  points  lying  on  the  line  joining  u 
and  v  or,  as  we  now  call  them,  A  and  B,  we  make  use  of  formulas  (7)  and 
(8)  developed  in  Chapter  II.  There,  in  order  to  locate  our  points,  we 
used  rectangular  coordinates  of  which  the  Y-axis  was  parallel  to,  and 
midway  between,  the  U-  and  V-axes  and  the  X-axis  was  the  line  joining 
the  zero  points  on  these  same  axes. 

The  formulas  for  the  coordinates  of  the  various  points  on  the  third 
line  of  the  diagram  were  then  found  to  be 

b  +  a         ~b+a' 
d  being  the  half  distance  between  the  U-  and  V-axes. 

In  these  equations  we  replace  a  by  \/#,  b  by  x2,  and  c  by  y. 

Now,  x  and  y  are  the  coordinates  either  of  the  points  representing 
the  observations  or  of  the  chosen  points  on  the  curve.  We  will  in  this 
instance  consider  them  as  belonging  to  points  on  the  curve. 


88 


CONSTRUCTION    OF   GRAPHICAL   CHARTS 


Below  are  tabulated  the  quantities  we  shall  require,  x  and  y  being 
read  from  the  curve  and  x  being  given  in  1000  pound  units: 


y 

X 

X* 

^ 

0.015 

0.15 

0.0225 

0-531 

0.0215 

0.5 

0.25 

0.794 

0.0284 

1.0 

1.0 

.0 

0.0392 

2.  O 

4.0 

.26 

0.05 

3-o 

9.0 

•  44 

o.  0616 

4.0 

16.0 

•59 

0.074 

5-o 

25.0 

•71 

0.0808 

5-5 

3°-  25 

•77 

Substituting  in  the  equations  for  X  and  Y  we  have  for  x  =0.15 


. 
X=  d 


0.0225  — 


0.0225  +  0.531 


—0.5085 
0-5535 


-0.9188  d, 


0.015 


#=0.5 
x=i.o 

X=2.O 
#  =  3.0 


0-5535 

X=  —0.521  d 
X  =  o.o 
X  =  o.$2i  d 
Jf =0.724  d 
d 


=  0.0271. 

7=o.o2o6o 
7=0.01420 


7=0.00479 
7=0.00350 
7=0.00277 
7=0.00252 

These  points  are  now  plotted  on  the  alinement  chart,  shown  in  Fig.  47. 
In  this  chart  the  distance  between  the  U-  and  V-axes  has  been  made  20; 
hence  d=io,  and  the  values  calculated  for  X  will  be  multiplied  by  this 
before  laying  them  off.  The  values  of  7  are  laid  off  to  any  convenient 
scale  which  will  give  clear  readings.  The  measurements  on  the  U-  and 
V-axes  are  to  this  scale. 

The  points  with  the  exception  of  the  first  two  seem  to  be  in  nearly 
perfect  alinement,  which  leads  us  to  infer  that  the  formula  chosen  is  a 
good  one.  If  they  fail  to  line  up  in  a  satisfactory  manner  it  is  useless  to 
go  further,  as  this  is  an  indication  that  the  wrong  equation  is  being  used. 
Of  course,  if  the  ordinates  taken  from  the  observations  themselves  had 
been  used  instead  of  the  points  on  the  curve,  we  could  not  expect  them 
to  fall  so  nearly  on  a  straight  line,  but  they  should  be  grouped  close  enough 
to  one  to  make  it  evident  that  the  axis  of  the  group  is  straight  and  not 
curved. 


EMPIRICAL  EQUATIONS 


The  line  extended  cuts  the  U-axis  at  0.0274  and  the  V-axis  at  o.oon, 
which  are,  therefore,  the  desired  values  of  A  and  B.  Before  using  them 
in  the  equation,  however,  we  shall  have  to  modify  them  slightly  to  take 
account  of  the  change  in  size  of  the  pressure  unit  which  is  really  1000 
times-  that  which  we  have  been  working  with.  Thus  A  will  have  to  be 
divided  by  ^/iooo,  or  10,  and  B  will  have  to  be  divided  by  iooo2,  or 
1,000,000,  and  our  final  formula  becomes,  after  substituting  tjd  for  y 
and  P  for  x, 

ifd  —  0.00274  *$/P+ o.oooooooo 1 1  P2. 

The  formula  was  now  solved  for  a  series  of  values  of  P,  and  the  results 
are  shown  by  the  double  circles  on  the  chart.  The  curve  could  not  be 
drawn  in  a  satisfactory  manner  as  it 
lies  very  close  to  the  original  for  a 
considerable  portion  of  its  length, 
and  this  closeness  is  a  good  indica- 
tion of  the  success  of  the  method. 

Lest  I  be  misunderstood,  let  me 
say  here  that  I  make  no  pretense  at 
having  obtained  a  better  mathe- 
matical expression  for  his  results 
than  Professor  Stewart.  The  scarcity 
of  data  in  the  region  of  higher  values 
renders  it  extremely  unsafe  to  say 
whether  the  line  there  is  straight  or 
curved.  What  interested  me  mainly 
in  this  problem  was  the  possibility  of  ,, 

'  FIG.  47.— Almement  diagram  for  testing 

expressing  the  entire  series  of  results     points  found  in  determining  equation  for 

by  one  formula.     This,  I  believe,  has    curve  of  Fig'  46' 

been  accomplished  with  a  very  fair  degree  of  success  and  by  the  use  of  a 

comparatively  simple  equation. 

The  method  we  have  been  investigating  is  generally  quite  sensitive, 
and  if  the  equation  is  not  a  good  one  for  the  purpose  the  points  will  depart 
markedly  from  the  straight  line.  Thus  the  possibility  of  forcing  an  un- 
suitable equation  into  the  appearance  of  an  agreement  with  the  original 
curve,  which  may  be  done  with  most  of  the  other  methods,  is  largely 
absent  here.  Often  a  portion  of  the  points  will  lie  along  a  straight  line 
while  the  others  depart  from  it.  In  this  case  it  indicates  that  in  a  limited 
field  the  compensation  is  possible  and  may  be  good,  a  fact  which  it  is 
sometimes  desirable  to  ascertain. 

In  the  example  just  explained,  we  have  assumed  that  not  only  the 


QO  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

general  type  of  the  compensating  equation  was  known,  but  also  the 
values  of  the  exponents  of  x. 

Some  guide  to  the  choice  of  the  exponents  is  evidently  much  to  be 
desired,  since  if  we  rely  upon  guesswork  we  may  consume  a  great  deal  of 
valuable  time  in  hunting  for  them,  and  may  even  then  not  hit  upon  the 
best  values. 

ANOTHER  ILLUSTRATION  OF  THE  ALINEMENT-CHART  METHOD. 

The  Batailler  method  just  described  may  be  extended  to  do  this  for 
many  types  of  equation  in  a  manner  which  is  comparatively  simple  in 
operation,  though  a  little  difficult  to  explain.  The  additional  one  or  two 
constants  which  may  thus  be  determined  are  not  limited  to  exponents, 
but  may  also  be  coefficients. 

As  before,  I  shall  make  the  explanation  while  working  out  a  problem. 
The  example  chosen  will  be  taken  from  Rateau's  "Flow  of  Steam  Through 
Nozzles,"  and  is  the  diagram  shown  in  Plate  IV  of  that  book  for  Hirn's 
experiments  on  the  flow  of  air  through  thin  plate  orifices.  I  have  redrawn 
the  curve  for  this  in  Fig.  48.  In  it  the  abscissas  represent  the  ratio  of 
back  pressure  p  to  initial  pressure  P,  and  the  ordinates  the  ratio  of 
observed  discharge  to  the  maximum  discharge.  The  abscissas  on  the 
X-axis  are  numbered  from  i  to  0.4,  but  I  have  reversed  the  numbering 
in  order  to  avoid  confusion  and  will  afterward  insert  the  quantities  as 
they  appear  in  the  original  diagram.  My  numbers  will  then  be  o,  o.i, 
0.2,  0.3,  etc.,  instead  of  i,  0.9,  0.8,  0.7,  etc. 

The  problem  will  be  to  see  how  nearly  we  can  compensate  this  curve 
by  an  equation  of  the  type 

y  =  Axp  +  Bxq 
in  which  A ,  B,  p,  and  q  are  all  unknown  and  are  to  be  evaluated. 

The  first  step  is  to  differentiate  the  curve  and  obtain  its  first  and 
second  derivatives,  y'  and  /'.  Then  A  and  B  are  eliminated  from  these 
equations,  and  p  and  q  determined  by  a  process  analogous  to  that  last 
described. 

The  equation  and  its  first  and  second  derivatives  are : 

y  =  Axp  +  Bxq 

y'  =  Apxp~l  +  Bqxq~l 

y"  =  Ap(p  -  i)xp~2  +  Bq(q-i)xq~2 

To  eliminate  A  and  B  from  these  equations  and  put  them  in  the  neces- 
sary form  for  use,  I  am  constrained  to  use  determinants.  Any  other 
method  lands  us  in  such  a  snarl  of  equations  as  to  be  very  objectionable, 


Ratio  of  Back  ITes.nre  p  to  Initial  Pressure  P. 


FIG.  48. — Chart  showing  relation  of  ratio  of  back  pressure  to  initial  pressure,  and  ratio 
of  observed  to  maximum  discharge  of  air  through  thin  plate  orifices. 


CONSTRUCTION  OF  GRAPHICAL  CHARTS 


while  by  determinants  we  can  reach  the  desired  results  by  a  com- 
paratively simple  process.  All  forms  of  equation  will  not  demand  this 
treatment,  and  each  case  must  be  looked  upon  as  more  or  less  of  a 
special  problem. 

The  three  equations  may  be  written  in  the  determinant  form  as 
follows: 

Axp  Bxq 


y"        Ap(p-i)xp~2        Bq(q-i)xq~2 

Divide  the  second  column  by  Axp  and  the  third  column  by  Bxq  and 
we  have: 


qx 


f 


=o 


Then  multiply  the  second  row  by  x  and  the  third  by  x2,  giving  : 
y  i  i 


xy 


p 


=o 


Taking  the  three  columns  as  the  coefficients  of  three  equations  of 
the  alinement  type,  we  have: 

y=xy'u  +x2y"v 
i=pu+p(p-i)v 
i  =  qu+q(q-i)v 

The  first  equation  is  affected  only  by  x  or  its  functions  y,  y',  and  y", 
the  second  by  p  only,  and  the  third  by  q  only. 

Three  supports  for  an  alinement  diagram  may  be  constructed  from 
these  three  equations  on  the  same  U-  and  V-axes,  giving  us  three  curves, 
one  for  x,  one  for  p,  and  one  for  q. 

If  we  join  up  some  point  on  the  p-line  with  another  on  q,  the  con- 
necting line  will  cut  the  #-line  in  what  must  be  looked  upon  as  a  corre- 
sponding value.  But  according  to  the  original  assumption  p  and  q  were 
constants  in  the  equation  and  remain  so  whatever  the  value  of  x.  If 
this  is  true,  the  desired  values  of  p  and  q  must  be  so  located  that  the  line 
joining  them  will  cut  every  value  of  x  on  the  x-\me.  This  can  only  be 
possible  by  having  the  support  for  x  a  straight  line  joining  these  constant 
values  of  p  and  q. 

Our  next  step  is  to  plot  the  support  for  x  from  the  equation: 

y=xy'u+x2y"v 


EMPIRICAL   EQUATIONS  93 

The  coordinates  for  the  points  on  this  line  will  be  found  from  the 
equations  used  in  the  previous  example,  which  will  read  here 


V_ 


Y=  — 

Here  x  and  y  are,  of  course,  the  coordinates  of  any  points  on  the  obser- 
vation curve;  y'  and  y"  must,  however,  be  determined  from  this  primary 
curve.  As  is  well  known,  the  tangent  to  a  curve  at  any  point  corresponds 
to  the  first  derivative.  If  we  get  the  tangents  at  a  sufficient  number  of 
points  their  values  may  be  plotted  into  a  second  curve  of  which  the  ordi- 
nates are  y'.  Similarly  by  drawing  tangents  to  this  second  curve,  we  get 
the  values  of  the  quantity  y". 

These  values  of  y'  and  y"  are  then  to  be  substituted  in  the  equations 
for  X  and  F. 

The  chief  and  only  difficulty  connected  with  this  process  is  in  drawing 
the  tangents  to  the  curves.  The  "curve  of  error"  is  sometimes  recom- 
mended for  this  purpose  but  is,  in  my  opinion,  too  cumbersome  for  prac- 
tical use  where  any  considerable  number  of  points  is  to  be  operated  on. 

My  own  preference  is  for  taking  two  ordinates  at  equal  distances  on 
either  side  of  the  point  at  which  the  tangent  is  desired  and  draw  the  chord 
of  the  curve  between  them.  If  the  curve  is  flat  these  side  ordinates  may 
be  considerably  separated,  but  if  not  they  must  be  closer.  The  slope 
of  the  chord  will  be  nearly  equal  to  that  of  the  tangent.  The  greatest  care 
must  be  exercised  in  this  part  of  the  process,  but  if  this  is  done  the  method 
will  yield  results  of  a  very  satisfactory  character. 

To  get  the  numerical  value  of  the  tangent,  or  y',  a  parallel  to  the  chord 
is  drawn  from  a  point  on  the  base  line  at  unit  distance  to  the  left  of  the 
foot  of  the  ordinate  we  are  operating  on  and  its  intersection  with  the 
ordinate  gives  the  desired  value  of  yf  when  measured  by  the  same  scale 
as  was  used  for  the  ordinates  of  the  original  curve. 

If  this  unit  distance  is  inconveniently  small  or  large  we  may  increase 
or  diminish  it  to  a  more  suitable  value  but  must  remember  that  the  read- 
ing on  the  ordinates  must  then  be  changed  to  correspond. 

In  Fig.  48  the  primary  curve  and  its  first  and  second  derivatives  are 
shown,  the  last  being  represented  only  by  points  and  the  actual  curve 
omitted  as  unnecessary  .  It  was  convenient  in  laying  out  these  secondary 
curves  to  use  a  base  unit  on  the  X-axis  equal  to  i  /io;  hence  the  readings 
on  the  yf  curve  must  be  multiplied  by  io  to  get  their  true  value  and  those 
of  the  y"  curve  by  100  (since  we  have  used  the  i/io  base  unit  twice). 


94 


CONSTRUCTION  OF  GRAPHICAL  CHARTS 


In  the  accompanying  table  are  given  the  quantities  necessary  for  our 
calculations,  the  values  of  y,  yf  and  y"  being  read  directly  from  the  curves; 
/',  it  will  be  noted,  has  the  minus  sign  prefixed  throughout,  as  its  points 
all  lie  below  the  X-axis. 


X 

y 

y' 

/' 

X2 

*y 

*/ 

o.  05 

0.283 

3-oo 

—34.5 

o.  0025 

—0.0863 

o.  150 

O.  10 

0.402 

1.94 

—  ii.  8 

0.0100 

—  0.1180 

0.194 

0.1$ 

0.488 

*-5° 

-  7-1 

0.0225 

—0.1598 

o.  225 

o.  20 

0-556 

1.19 

—  5-° 

o  .  0400 

—  0  .  200O 

0.238 

0.25 

0.613 

I  .  00 

-  3-7 

o.  0625 

—0.2313 

o.  250 

0.30 

0.658 

0.83 

-  2.85 

o  .  0900 

—0.2565 

0.249 

o-35 

0.695 

o.  70 

-  2.35 

o.  1225 

—  o.  2879 

0.245 

0.40 

0.728 

0.605 

—  2.  O 

o.  1600 

—  0.3200 

o.  242 

0.45 

Q-757 

o.  50 

—  i-9 

o.  2025 

-0.3848 

o.  225 

o.  50 

o.  780 

0.42 

-  i-7 

o.  2500 

—0.4250 

0.210 

°-55 

o.  800 

0.33 

—  1.6 

0.3025 

—  o  .  4840 

o.  1815 

Now,  substituting  from  the  first  row  in  the  equations  given  above  for 
X  and  F,  we  have 

^=^-o.o863-_ai5  =  ^-o.2363         m  _  ^ 


-0.0863  +  0.15 

„     0.283 


0.0637 
and  for  the  remaining  values  of  x, 


0.0637 
=  4-45> 


OC  =  O.IO 

X=  —  4.  nd 

y=  5-3 

#=0.15 

X=~   $.gid 

Y-   7-5 

#  =  O.2O 

X=-n.$$d 

K=i4.62 

#  =  0.25 

X=-2$.7$d 

F  =  32.8 

#  =  0.30 

^  =  67.5^ 

F=-87.8 

*  =  0.35 

X=i2.45^ 

F=-i6.2 

x  =  o  .  40 

X=   j.2id 

y=~  9-35 

#=0.45 

X=  3.8i<2 

F=-  4-74 

#  =  0.50 

X=   2.96d 

y—  3-63 

^  =  0.55 

X=   2.igd 

F=-  2.64 

Slide-rule  calculations  are  usually  of  sufficient  accuracy  for  this  pur- 
pose, and  after  a  start  is  once  made  they  may  be  run  off  quite  rapidly. 

These  values  must  next  be  plotted  on  the  alinement  diagram,  as  shown 
in  Fig.  49.  Since  we  make  no  use  of  the  U-  and  V-axes  here,  we  will  omit 
them  and  indicate  only  the  X-  and  Y-axes  from  which  the  above  quan- 


EMPIRICAL   EQUATIONS 


95 


titles  are  laid  off.  The  half  distance  d  between  the  U-  and  V-axes  appears 
in  the  equation  for  X,  but  since  they  are  not  shown,  its  only  function  will 
be  that  of  a  scale  unit,  which  we  may  make  any  size  we  please.  Here  it 


1-32.8 


FIG.  49. — Alinement  diagram  for  testing  points  found  in  determining  an  equation 

for  curve  of  Fig.  48. 

was  made  of  such  a  size  that  all  of  the  points  given  above  could  be  plotted 
within  the  limits  of  Fig.  49,  except  the  one  corresponding  to  x  =  0.30. 
We  do  not  need  this  point,  however,  as  there  are  enough  other  points 
without  it  to  determine  the  alinement. 


96  CONSTRUCTION    OF   GRAPHICAL   CHARTS 

An  examination  of  Fig.  49  shows  that,  although  the  plotted  points  do 
not  lie  exactly  on  any  straight  line,  they  are  in  very  close  agreement  with 
the  one  shown.  Exact  agreement  is  never  expected,  of  course,  and  there 
will  generally  be  more  divergence  than  shown  here.  The  alinement  of 
the  points  indicates  that  the  type  of  equation  chosen  is  good  for  the  pur- 
pose. If  it  had  not  been  the  points  would  have  scattered  badly,  or  would 
have  had  a  curve  as  their  locus. 

Now,  we  must  plot  the  curves  for  p  and  q  for  such  values  of  these 
quantities  as  we  suppose  to  lie  near  the  line  just  drawn. 

The  p  and  q  equations  for  this  type  of  formula  are  identical,  hence 
they  will  be  represented  by  only  one  curve  instead  of  by  two,  as  is  the 
case  with  other  types  where  p  and  q  are  not  symmetrically  disposed.  In 
order  to  get  a  value  each  for  p  and  q,  we  must,  therefore,  have  two  inter- 
sections between  the  oc  support  and  the  suppport  for  p  q. 

The  alinement  equation  for  p  as  given  previously  is 
i=p  u  +  p  (p  —  i)  v. 

The  coordinates  for  the  various  points  on  the  p  support  will  then  be 
p(p-i)-p     ,f-ap_ 


These  equations  have  been  solved  for  p  =  0.5,  0.6,  0.7,  0.8,  0.9,  i,  1.5, 
and  2,  and  the  points  thus  found  plotted  in  Fig.  49. 

A  larger  scale  for  the  drawing  would  have  made  things  clearer,  but 
it  can  be  seen  that  the  ^-support  cuts  this  curve  at  two  points,  one  of  them 
exactly  at  1.5  and  the  other  at  a  point  between  0.5  and  0.6,  which  I  have 
called  0.53.  It  would  have  required  but  a  slight  shift  of  the  ^-support 
to  have  made  the  intersections  at  the  points  0.5  and  2.  If  convenience 
in  use  were  an  important  factor  these  latter  values  could  probably  be 
employed  with  but  little  less  accuracy.  However,  we  will  use  the  original 
more  exact  figures  and  call  p  1.5  and  #0.53.  It  is  immaterial  at  this  stage 
which  quantity  is  assigned  to  which  letter. 

Having  found  p  and  q,  our  equation  now  reads 


and  our  next  step  is  to  proceed  exactly  as  we  did  in  the  previous  example 
to  find  A  and  B,  using  the  equations 


v= 


in  order  to  locate  the  points  on  the  test  line. 


EMPIRICAL  EQUATIONS 


97 


I5U 


\.ttb 

i.n 

1.15 

i.oa 

1.01S 

O.'JS 

O.Mt 


As  there  is  nothing  novel  in  the  process  the  details  will  be  omitted 
and  reference  merely  made  to  Fig.  50,  where  the  points  have  been  plotted 
and  where  they  are  seen  with  but  one  exception  to  lie  almost  exactly  upon 
a  straight  line,  again  indicating  the  adaptability  of  the  formula  to  the  por- 
tion of  the  curve  we  have  operated  on.  This  line  extended  to  the  U- 
and  V-axes  is  found  to  cut  them 
at  the  points  —0.59  on  U  and 
+ 1.425  on  V. 

Then  A  is  —0.59  and  B,  1.425, 
and  our  equation  reads 

y=  -0.59  xl'*+  1.425  *°'53. 

Reverting  now  to  the  first  dia- 
gram where,  as  was  stated,  the 
numbering  on  the  X-axis  was 
altered,  we  see  that  we  can  get 
the  original  numbers  by  putting 

50. — Alinement  diagram  for  testing 
i)  IP  being  the  ratio  of  the  back  points  found  in  determining  an  equation  for 
^'  .  ...  .  curve  of  Fig.  48. 

to   the  initial  pressure  and  the 
final  equation  may  then  be  written 

y=  —0.59  (i—  plP)l'*+  1.425  (i—  />/P)°'53. 

A  series  of  points  has  been  calculated  from  this  formula  and  plotted 
on  the  chart  in  Fig.  48,  as  indicated  by  the  small  circles,  and  the  agree- 
ment will  be  seen  to  be  quite  good. 

The  lengthy  description  which  I  have  given  of  the  Batailler  method 
has,  I  know,  a  somewhat  formidable  sound,  but  in  practical  operation  "  It 
is,"  as  Bill  Nye  observed  of  Wagner's  music,  "much  better  than  it  sounds." 

The  only  operation  which  presents  much  difficulty  is  the  graphical 
differentiation  which  must  be  done  with  great  care,  or  the  results  will  be 
poor.  Otherwise  the  work  is  all  of  a  simple  character,  and  may  be  carried 
out  very  expeditiously  as  compared  with  some  of  the  other  processes. 
While  other  types  of  equation  are  developed  on  the  same  general  lines  as 
the  one  explained,  there  are  differences  of  detail  which  have  to  be  looked 
out  for,  and  which  could  not  even  be  touched  upon  here  without  lengthen- 
ing this  discussion  beyond  reasonable  limits.  The  equation  we  have 
worked  on  is,  perhaps,  the  one  most  commonly  met  with  in  practice  and 
shows  as  well  as  any  other  the  very  decided  advantage  of  this  process  for 
certain  classes  of  work. 


CHAPTER  VII. 

STEREOGRAPHIC  CHARTS  AND  SOLID  MODELS. 
THREE  DIMENSIONAL  CHARTS. 

Two  dimensional  charts  for  the  representation  of  mathematical  equa- 
tions or  experimental  data  are  in  very  common  use  nowadays  and  are 
everywhere  recognized  as  valuable  devices  for  giving  a  clear  conception 
of  the  manner  in  which  the  variables  are  related. 

Their  application  is  generally  restricted,  however,  to  cases  where  there 
is  but  one  variable  and  its  function,  if  the  variation  to  be  shown  is  continu- 
ous. Nevertheless  cases  often  arise  in  which  there  are  two  variables  and 
a  function  to  be  represented  and  where  it  is  desirable  to  show  a  continuous 
variation  for  all  three. 

A  simple  and  logical  extension  of  the  two-dimensional  chart,  in  which 
the  variation  is  represented  by  a  plane  curve,  leads  us  to  the  idea  of  a  solid, 
three-dimensional  chart  in  which  the  variation  is  shown  by  a  surface. 

It  has  received  some  attention  at  the  hands  of  a  number  of  writers  on 
engineering  matters  and  graphics,  but  for  some  reason,  probably  the  labor 
and  expense  involved  in  its  construction,  its  actual  use  has  been  rather 
limited.  Where  it  has  been  used  it  has  in  some  instances  been  fruitful 
in  good  results  and  has  thrown  much  light  upon  obscure  phenomena.  In 
this  connection  its  chief  value  has  probably  come  from  the  facility  with 
which  we  are  able  to  detect  maximum  and  minimum  conditions  and  rates 
of  change  among  variables  whose  relationship  is  complex  or  unknown. 
Often  we  must  deal  with  conditions  where  no  known  equations  will 
connect  our  experimental  results  and  where  a  mere  tabulation  of  figures 
will  not  yield  the  desired  information  without  much  tedious  study.  The 
well  recognized  superiority  of  any  graphical  representation  over  an  equa- 
tion or  table  in  conveying  a  clear  impression  to  the  mind  of  the  way  in 
which  a  set  of  variables  is  related  will  often  in  itself  be  a  sufficient  justifica- 
tion for  the  use  of  this  type  of  chart. 

Between  the  solid  model  and  the  plane  chart  there  is  a  borderland 
occupied  by  types  which  do  not  truly  belong  to  either  and  which  are 
really  plane  projections  of  solid  models.  They  may  be  orthographic, 
isometric,  perspective  or,  generally,  axonometric,  according  to  the  taste 
of  the  maker  or  the  exigencies  of  the  subject. 

The  orthographic  projection  here  referred  to  is  the  topographic  map 
projection  in  which  the  relief  of  the  model  is  indicated  by  a  series  of 


STEREOGRAPHIC   CHARTS  AND   SOLID   MODELS 


99 


contour  lines.  Each  line  passes  through  a  series  of  points  at  the  same 
elevation  and  is  numbered  to  show  this  elevation.  Only  a  slight  effort  of 
the  imagination  is  required  to  give  a  very  good  idea  of  the  undulating  sur- 
face which  they  represent.  The  familiar  weather  map  is  a  good 
example  of  such  a  chart.  Here  points  of  equal  barometric  pressure  are 
connected  by  curved  lines  called  isobars.  Charts  of  this  description  have 
been  much  used  to  record  tidal 
phenomena,  magnetic  observa- 
tions, etc.,  and  also  in  the  pre- 
sentation of  vital  and  financial 
statistics. 

Axonometric  projections  will 
usually  be  found  superior  to  the 
topographic  in  bringing  out  clearly 
the  shape  of  the  surface  and  are 
not  at  all  difficult  to  construct. 
The  special  case  where  the  pro- 
jection is  isometric  was  very  fully 
dealt  with  by  Prof.  Guido  Marx  in 
the  American  Machinist,  Volume 
31,  Part  2,  page  701. 

Any  of  the  other  well-known  methods  of  rectangular  axonometry  or  of 
perspective  may,  of  course,  be  applied  to  these  figures.  As  these  methods 
are  generally  understood  or  may  be  found  described  in  almost  any  good 
book  on  projection  or  descriptive  geometry,  no  attempt  will  be  made 
to  discuss  their  principles  here. 

The  accompanying  table  may,  however,  be  convenient  for  reference 
as  indicating  the  proper  choice  of  angles  for  the  axes  to  conform  to  the 
scale  units  most  commonly  used. 

TABLE   OF  RATIO   OF  UNIT  LENGTHS   ON  THE  AXES  AND  ANGLES  OF 
THE  AXES  FOR  AXONOMETRIC  PROJECTIONS 


FIG.  51. — Axes  and  angles  for  axonometric 
projection. 


Ratio  of  unit  lengths 

ux  :  uy  :  uz 

TVw.  <£ 

Tan.  0 

fi:.    ,1 

=  60° 

|   Isome  ric   J 

9 

2:1         2 

8  : 

.8:7 

3  :  i        3 

18  : 

18  :  17 

4:1        4 

32  : 

32  :  31 

5:4        6 

5  : 

3  :     i 

9:5      10 

ii  : 

25  :    8 

IOO  CONSTRUCTION    OF    GRAPHICAL   CHARTS 

The  letters  in  the  table  refer  to  the  same  symbols  in  Fig.  51  and  the 
scale  values,  designated  by  u,  represent  the  ratio  of  sizes  for  a  unit  length 
on  each  of  the  axes. 

The  question  of  scales  in  the  projection  of  such  figures  as  we  are  now 
considering  is,  however,  of  relatively  little  importance  since  the  units  used 
on  the  different  axes  have  generally  no  relation  which  makes  any  special 
scale  ratio  necessary.  The  angles  for  the  axes,  on  the  other  hand,  should 
be  so  chosen  as  to  agree,  approximately  at  least,  with  those  given  in  the 
table,  otherwise  the  figure  may  have  an  awkward  and  unnatural  ap- 
pearance. 

AXONOMETRIC   CHARTS. 

But  one  simple  illustration  will  be  given  for  this  type  of  chart  which 
will,  however,  show  some  interesting  and  rather  unusual  features.  It  is 
taken  from  the  Zeitschrift  des  Vereines  Deutscher  Ingenieure  for  December 
27,  1902,  and  occurs  in  an  article  by  O.  Lasche  on  the  friction  of  journals 
with  high  surface  velocities. 

Fig.  52  was  redrawn  from  a  chart  given  in  this  paper  with  a  few 
unimportant  modifications  to  render  it  better  adapted  to  purposes  of 
illustration.  The  chart  was  constructed  from  data  obtained  from  experi- 
ments on  a  nickel-steel  journal  running  in  a  white-metal  bearing  and  is 
intended  to  show  the  relation  between  the  temperature  of  the  bearing  in 
degrees  Centigrade,  the  surface  velocity  in  meters  per  second  and  the 
heat  generated  per  square  centimeter  of  effective  projected  area,  expressed 
in  heat  units,  and  also  in  meter-kilograms  per  second. 

The  experiments  were  made  at  a  specific  pressure  of  6.5  kilograms  per 
square  centimeter,  but  since,  with  the  lubrication  used,  the  product  of  the 
specific  pressure  and  the  coefficient  of  friction  was  sensibly  constant  over 
a  considerable  range,  the  results  are  said  to  be  applicable  to  any  specific 
pressure  from  3  to  1 5  kilograms  per  square  centimeter. 

In  laying  out  a  chart  of  this  description  the  three  coordinate  axes  and 
then*  planes  are  first  drawn  and  the  former  properly  graduated  between 
the  limits  set  by  the  experiments.  From  the  graduations  on  the  ground 
plane  axes  perpendiculars,  lying  in  the  ground  plane,  are  drawn,  thus 
giving  a  checkered  surface  on  which  points  may  be  located  as  is  done  with 
ordinary  section  paper.  At  the  points  thus  found  perpendiculars  are 
erected  to  the  ground  plane,  their  height  being  so  taken  as  to  represent 
the  value  of  the  third  variable.  The  tops  of  these  lines  are  now  connected 
by  suitable  curves,  which  must  lie  in  the  surface  we  are  seeking  and  which 
are  assumed  to  represent  it. 


STEREOGRAPHIC   CHARTS  AND    SOLlD   MODELS 


In  the  chart  under  discussion  five  different  curves  were  drawn  parallel 
to  the  temperature-heat  plane,  and  then,  to  bind  them  together  and 
render  the  shape  of  the  surface  more  apparent,  three  more  curves  were 
drawn  at  right  angles  to  the  first  and  parallel  to  the  velocity-heat  plane. 
Taking  one  of  these  latter  curves,  that  corresponding  to  50  degrees,  we  see 
that  at  this  constant  temperature  the  heat  generated  by  friction  increases 
with  increasing  velocity,  not  exactly  in  direct  ratio  with  it,  however,  since 


FIG.  52. — Axonometric  chart  showing  relation  between  journal  bearing  temperature, 
surface  velocity  and  heat  generated. 

the  coefficient  of  friction  does  not  remain  quite  constant  as  the  velocity 
changes.  Keeping  the  velocity  constant  and  varying  the  temperature, 
we  see  that  the  heat  generated  by  friction  decreases  as  the  temperature 
rises,  rapidly  at  first  and  then  more  slowly. 

It  is  evident  from  the  chart  that  with  the  journal  in  question  the  heat 
produced  by  friction  will  be  greatest  when  it  is  starting  up  and  the  tem- 
perature is  low.  Also  that  at  this  temperature  the  radiation  to  the 
surrounding  atmosphere  will  be  small  on  account  of  the  small  temperature 
difference.  The  heat  produced  therefore  goes  to  warm  the  bearing,  but 
as  its  temperature  rises  the  heat  generated  becomes  less  and  the  radiation 


CONSTRUCTION  OF  GRAPHICAL  CHARTS 


greater  until  we  reach  a  point  where  the  radiation  just  balances  the  heat 
production  and  the  temperature  remains  stationary. 

The  question  naturally  arises  as  to  whether  it  is  possible  to  tell  where 
this  point  will  be.  If  the  necessary  experimental  data  are  at  hand  it  may 
be  done  on  the  chart.  Suppose  we  have  this  data  and  from  it  construct  a 
second  chart  on  the  same  heat,  temperature  and  velocity  axes  as  before. 
See  Fig.  53.  It  shows  the  capacity  for  heat  radiation  per  square  centi- 
meter of  effective  projected  area  for  the  bearing  we  are  considering  and  is 


FIG.  53. — Axonometric  chart  showing  capacity  for  heat  radiation  per  unit  of 
effective  projected  area  of  journal  bearing. 

constructed  for  a  room  temperature  of  20  degrees  Centigrade.  When  the 
bearing  has  this  temperature  its  radiation  is,  of  course,  zero.  The  radiation 
is  independent  of  the  velocity  of  the  journal  and  this  is  indicated  by  the 
fact  that  the  surface  is  a  ruled  one  composed  of  straight  lines  parallel  to 
the  velocity  axis.  Increasing  bearing  temperature  means,  of  course, 
increasing  radiation. 

Next  suppose  these  two  charts  to  be  combined  as  in  Fig.  54.  It  is 
apparent  that  the  two  surfaces  will  intersect  along  some  line  as  h  c  j,  the 
location  of  which  is  easily  found  by  the  rules  governing  this  form  of  pro- 
jection. 


STEREOGRAPHIC   CHARTS  AND   SOLID   MODELS 


103 


Any  point  on  this  line  will  correspond  to  some  temperature  and  veloc- 
ity at  which  the  radiation  just  equals  the  heat  production,  the  necessary 
condition  for  constant  temperature.  This  line  projected  to  the  ground 
plane  gives  us  the  line  i  d  j.  Any  point  in  the  ground  plane,  projected 
from  the  temperature  and  velocity  axes,  which  falls  in  front  of  the  line 
will  indicate  that  under  these  conditions  the  radiation  is  greater  than  the 


,  •  •.„        FIG.  54. — Chart  combining  charts  of  Figs.  52  and  53. 

heat  generation  or  that  natural  cooling  will  be  effective  to  keep  the  bearing 
below  the  chosen  maximum  temperature.  Points  which  fall  beyond  this 
line  correspond  to  conditions  where  artificial  cooling  must  be  resorted  to. 
Suppose,  for  instance,  we  take  some  point,  such  as  c  on  the  line  h  c  j. 
Project  it  to  the  ground  and  we  find  that  it  falls  at  the  intersection  of  ordi- 
nates  from  80  degrees  on  the  temperature  axis  and  5  meters  per  second 


104  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

on  the  velocity  axis.  Under  these  conditions  the  temperature  will  be 
steady. 

Next  suppose  that  we  arbitrarily  fix  the  upper  limit  for  temperature 
at  80  degrees,  and  that  we  have  a  velocity  of  lo-meters  per  second. 

Entering  the  radiation  diagram  on  the  8o-degree  line  at  a  we  run  up 
till  we  reach  its  surface  at  b.  Then,  following  the  surface  along  the  line 
be,  we  find  its  intersection  with  the  ic-meter  plane  of  the  friction  diagram 
at  e.  Through  this  point  a  perpendicular  is  drawn  to  the  ground.  The 
length /#  on  this  perpendicular  measures  the  heat  generated  by  friction, 
efis  the  amount  carried  off  by  radiation,  while  eg  represents  the  remainder 
which  must  be  artificially  removed  by  circulating  a  current  of  water,  oil,  or 
air  around  the  bearing. 

The  writer  of  the  article  from  which  I  take  this  illustration  goes  on  to 
show  how,  after  measuring  eg  in  heat  units,  a  very  simple  calculation  will 
give  the  amount  of  cooling  fluid. 

It  will  be  apparent  from  the  foregoing  description  that  the  axono- 
metric  projection  has  some  advantages  over  its  solid  prototype  from  the 
facility  with  which  we  can  project  through  the  figure  in  case  of  need. 
Special  attention  should  also  be  directed  to  the  use  which  has  been  made 
of  the  line  of  intersection  of  the  two  surfaces.  It  is  a  rather  novel  feature 
and  one  which  should  prove  valuable  in  many  engineering  problems. 

THE  SOLID  MODEL. 

Next  let  us  consider  the  true  solid  model.  It  has  received  attention 
at  the  hands  of  several  eminent  writers,  among  them  the  late  R.  H.  Thur- 
ston.  He  published  a  number  of  articles  explaining  its  uses  and  advan- 
tages, among  which  articles  may  be  cited  one  on  glyptic  models  in  the 
Transactions,  American  Society  of  Mechanical  Engineers,  for  1898.  He 
appears  to  have  been  much  impressed  by  the  possibilities  it  offered  for  the 
solution  of  a  certain  class  of  problems  and  he  illustrates  its  application 
by  a  number  of  examples. 

In  spite  of  his  optimistic  views  as  to  its  value,  the  solid  model  has  never 
seemed  to  "take"  well;  at  least  there  are  relatively  few  recorded  instances 
of  its  use.  This  may  be  partly  due,  as  was  observed  before,  to  the  labor 
involved  in  its  construction,  but  possibly,  also,  to  a  lack  of  sufficient 
exploitation. 

These  models  may  be  made  in  various  ways.  Wood  is  a  suitable 
material  where  the  surface  to  be  produced  is  sufficiently  regular,  but  this 
is  not  often  the  case.  Ruled  surfaces  may  be  produced  by  strings 


STEREOGRAPHIC   CHARTS  AND    SOLID    MODELS 


105 


stretched  on  suitable  frames,  but  the  material  most  generally  used  is 
plaster  of  Paris.  After  the  first  model  is  made,  replicas  may,  of  course, 
be  cast  in  any  suitable  metal  or  material.  Cardboard,  as  will  be  shown 
later,  is  a  cheap  and  convenient  substitute  for  some  of  the  above-named 
materials. 

In  making  the  plaster-of-Paris  model,  we  first  stretch  a  sheet  of  section 
paper  on  a  board  and  lay  off  on  it  the  points  corresponding  to  two  of  the 
variables  in  the  usual  manner.  At  these  points  we  next  insert  vertical 


FIG.  55. — Solid  model  showing  relation  between  heat  units  per  hour  per  brake  horse- 
power, compression  pressure  and  volume  of  gas  mixture  for  a  gas  engine. 

wires  which  are  cut  off  at  heights  corresponding  to  the  third  variable.  A 
box  is  then  formed  around  the  whole  and  wet  plaster  of  Paris  is  poured 
into  it  until  all  the  wires  are  covered.  After  it  has  set,  the  upper  surface 
is  carefully  cut  and  smoothed  away  until  the  tops  of  the  wires  are  exposed 
and  the  resulting  surface  is  taken  as  the  graphical  representation  of  the 
law  of  connecting  the  variables. 

A  comparatively  recent  example  of  such  a  model  is  found  in  an  article 
on  the  mixture  ratio  for  gas  engines  in  the  Zeitschrift  des  Vereines  Deut- 
scher  Ingenieure  for  September  14,  1907. 

This  model  is  represented  in  Fig.  55.  It  is  based  on  data  obtained 
from  a  gas  engine  running  at  four  horsepower  on  producer  gas,  and  is 
intended  to  show  the  relation  between  the  heat  units  per  hour  per  brake 


106  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

horsepower,  the  compression  pressure  in  atmospheres  and  the  volume  of 
mixed  gas  and  air  per  1000  heat  units  of  lower  heating  value. 

The  front  horizontal  axis  is  graduated  to  represent  the  cubic  meters 
of  mixture  per  1000  heat  units  and  extends  from  about  1.7  cubic  meters 
to  3."  Perpendicular  to  this  axis,  and  also  horizontal,  is  the  axis  for  com- 
pression pressures  graduated  from  back  to  front  between  4  and  13 
atmospheres. 

The  vertical  axis  is  used  for  the  heat  units  required  per  brake  horse- 
power-hour, the  graduations  beginning  at  2500  at  the  ground  plane  and 
running  up  to  5000. 

The  hollowed  surface  in  the  middle  of  the  model  covers  the  range 
within  which  the  experiments  were  conducted,  and  the  cut-off  portions 
at  the  sides  of  the  hollow  have  no  meaning. 

Without  in  any  way  attempting  to  discuss  the  conditions  under  which 
the  experimental  results  were  obtained,  we  will  take  the  model  as  it 
stands,  and  see  what  conclusions  may  be  reached  from  a  simple 
inspection  of  it. 

The  bottom  of  the  valley  indicates  the  lowest  heat  consumption  per 
horsepower-hour  in  any  given  locality. 

The  intersection  of  the  valley  with  the  back  vertical  plane  is  a  curve 
somewhat  resembling  a  parabola  with  steeply  rising  sides.  As  we  come 
toward  the  front  the  curves  cut  by  parallel  vertical  planes  flatten  out  and 
the  vertex  of  the  curve  becomes  lower,  indicating  a  smaller  heat  consump- 
tion as  the  compression  increases.  At  the  back  of  the  model  the  lowest 
part  of  the  curve  is  tangent  to  a  horizontal  line  at  about  4100,  while  in 
front  it  touches  3100. 

It  will  also  be  noted  that  the  slope  of  the  bottom  of  the  valley  is  steepest 
in  the  rear  and  is  nearly  horizontal  in  front,  indicating  a  more  rapid  gain 
in  heat  economy  from  increased  compression  when  the  original  compres- 
sion is  low  than  when  high. 

The  bottom  of  the  valley  shows  a  tendency  to  drift  to  the  left  as  we 
come  forward,  indicating  that  with  increased  compression  the  best  econ- 
omy was  obtained  by  increasing  the  dilution  of  the  mixture.  The  flatten- 
ing out  of  the  front  part  of  the  valley  indicates  that  as  compression  in- 
creases the  necessity  for  an  exact  mixture  ratio  for  good  economy  becomes 
progressively  less  important. 

These  points  are  all  interesting,  and  while  they  might  have  been  dis- 
covered from  an  inspection  of  a  series  of  curves  or  of  the  tabulated  data, 
it  is  clear  that  the  model  has  greatly  simplified  the  process  of  deduction, 
and  has  thus  justified  its  construction. 


STEREOGRAPHIC   CHARTS   AND    SOLID   MODELS 


107 


CARDBOARD  SUBSTITUTE  FOR  THE  SOLID  MODEL. 

Reference  has  been  made  above  to  the  cardboard  model  as  a  cheap 
substitute  for  the  solid  type.  The  next  illustration  will  be  an  example 
showing  its  construction. 

If  we  assume  one  of  three  variables  to  have  different  constant  values, 
we  get  a  series  of  plane  curves  connecting  the  other  two.  Then,  by  doing 
the  same  with  one  of  the  other  variables,  we  get  a  second  series  of  curves 


«  FIG.  56. — Cardboard  substitute  for  a  solid  model. 

for  planes  at  right  angles  to  the  first.  Each  of  these  curves  is  cut  from 
a  piece  of  cardboard  and  slit  half  way  up  or  down  the  lines  of  intersection 
with  the  cards  at  right  angles  to  it.  They  are  then  fitted  together  some- 
thing on  the  principle  of  an  egg  box,  and  the  result  will  be  a  series  of  plane 
curves,  properly  spaced  with  reference  to  each  other,  all  of  which  lie  in 
the  surface  we  are  trying  to  represent.  It  is  evidently  closely  related  to 
the  axonometric  projection  previously  described. 

Such  a  model  is  shown  in  Fig.  56.  It  was  constructed  from  the  curves 
given  in  a  paper  in  the  Transactions,  American  Society  of  Mechanical 
Engineers,  for  1904,  by  E.  S.  Farwell,  entitled  "Tests  of  a  Direct  Con- 
nected Eight  Foot  Fan  and  Engine."  These  curves  were  chosen  chiefly 
on  account  of  the  irregular  hilly  character  of  the  surface  to  which  they 
belong  as  affording  a  good  test  of  the  method.  They  occur  in  Fig.  39,  of 
the  article  referred  to,  and  are  supposed  to  show  the  relation  between  the 


108  CONSTRUCTION  OF  GRAPHICAL  CHARTS 

efficiency  of  the  fan  and  the  area  of  the  outlet  opening  for  various  speeds, 
the  area  of  opening  being  designated  as  a  percentage  of  the  product  of  the 
fan  diameter  by  the  width  of  periphery. 

Eight  different  curves  are  given  for  eight  different  speeds,  which 
advance  by  steps  of  25  from  50  revolutions  per  minute  to  225  revolutions 
per  minute.  The  curves  shown  in  the  figure  had  the  efficiencies  plotted 
as  ordinates,  and  the  outlet-opening  percentages  as  abscissas,  and  were 
used  as  they  stood  for  one  set  of  cards.  Then  taking  the  intersection  of 
these  curves  with  one  of  the  perpendiculars  to  the  outlet  axis  we  get  a 
series  of  lengths  which  we  use  as  equally  spaced  ordinates  for  a  curve  at 
right  angles  to  the  plane  of  the  original  drawing,  the  ordinates  again 
representing  efficiencies  while  the  abscissas  this  time  are  velocities.  As 
many  similar  curves  as  were  deemed  necessary  were  taken  from  the  other 
perpendiculars.  All  were  cut  out  and  fitted  together,  forming  the  model 
shown  in  the  photograph.  Such  a  model  may  be  applied  to  many,  if  not 
most,  of  the  purposes  for  which  the  solid  type  is  used  and  has  a  decided 
advantage  in  simplicity  of  construction. 

THE  TRI-AXIAL  MODEL. 

Before  leaving  this  subject  a  brief  reference  must  be  made  to  an  in- 
genious form  of  solid  chart  described  by  Professor  Thurston  in  several  of 
his  articles.  It  is  called  the  tri-axial  model.  By  its  use  it  is  possible  to 

take  into  account  four  different  variables 
instead  of  three  as  was  previously  the  case. 
It  is  a  necessary  condition,  however,  that 
for  each  set  of  corresponding  variables  three 
of  them  should  add  up  to  a  constant  value, 
generally  100  per  cent.  The  fourth  is  unre- 
stricted. These  models  have  been  very  use- 
ful in  representing  the  properties  of  ternary 
alloys,  furnace  slags,  etc.  If  we  have  an 
^uilateral  triangle  as  shown  in  Fig.  57,  and 
from  any  point,  O,  within  it  we  drop  per- 
pendiculars to  the  three  sides,  geometry  tells  us  that  the  sum  of  these 
perpendiculars  is  constant  wherever  the  point  may  be  located. 

Therefore,  if  we  wish  to  study  the  alloys  composed  of,  say,  copper, 
tin,  and  zinc,  with  reference -to  any  property  such  as  strength,  ductility, 
hardness,  or  melting  point,  a  large  number  of  experiments  are  made  with 
specimens  of  varying  composition  and  the  value  of  the  quality  we  are 
studying  tabulated  with  the  composition  of  the  alloy. 


STEREOGRAPHIC    CHARTS   AND    SOLID    MODELS 


This  composition  is  expressed  for  each  constituent  as  a  percentage, 
and  the  three  percentages  necessarily  add  up  to  100. 

Laying  out  a  triangle  whose  altitude  to  some  scale  is  100,  we  designate 
one  side  as  copper,  another  as  tin,  and  the  third  as  zinc.  Parallels  are 
then  drawn  to  the  sides,  distant  from  them  by  amounts  corresponding  to 
the  percentage  of  each  metal  in 
the  specimen.  The  scale  in  which 
these  distances  are  measured  is 
the  same  as  that  which  was  used 
in  laying  off  the  altitude  of  the 
triangle.  These  three  parallels 
must  meet  in  a  point  which  is 
taken  to  represent  the  alloy  in 
question.  Perpendicular  to  the 
ground  plane  at  this  point  we  in- 
sert a  wire  whose  length  repre- 
sents the  value  of  the  quality  we 
are  studying.  When  all  the  wires 
are  fixed  the  whole  is  covered  with 
plaster  of  Paris,  as  explained  be- 
fore,  which  is  then  pared  down  to 
the  tops  of  the  wires. 

The  resulting  model  is  shown  in  Fig.  58,  and  from  it  Thurston  found 
that  the  strongest  alloy  had  a  composition  of  Cu  =  55  per  cent.,  Zn  =  43 
per  cent.,  and  Sn  =  2  per  cent. 

Models  of  this  description  are  evidently  of  especial  value  in  the  study 
of  metallurgical  problems  and  are  by  no  means  uncommon,  particularly 
in  that  field  of  work. 

Often,  however,  instead  of  the  solid  model,  a  topographical  chart  of 
it  with  the  necessary  contour  lines  is  plotted,  which  answers  many  pur- 
poses almost  equally  well  and  commends  itself  for  use  in  a  great  many 
cases. 


CHAPTER  VIII. 

THE  USE  OF  DETERMINANTS. 

The  labor  of  laying  out  charts  of  the  alinement  type  may  be  considerably 
simplified  and  abbreviated  if  use  is  made  of  determinants.  The  troublesome 
question  of  scale  units  is  more  easily  handled,  and  the  arrangement  of  the 
axes  is  under  more  intelligent  control  than  is  possible  by  the  methods  pre- 
viously described. 

The  actual  knowledge  of  determinants  required  for  this  purpose  is,  in 
the  great  majority  of  the  cases  likely  to  be  met  with,  extremely  small,  so 
small  indeed  that  I  shall  attempt  to  cover  all  of  the  really  essential  points 
within  the  few  pages  of  explanation  which  follow.  Anyone  with  an  elemen- 
tary knowledge  of  mathematics  should  be  able  to  understand  this  with  little 
difficulty,  and  the  superiority  of  the  method  will  amply  justify  the  small 
amount  of  labor  required  to  learn  the  use  of  a  new  tool. 

In  what  follows  there  will  be  little  beyond  a  bare  outline  of  the  essential 
facts,  and  if  some  of  my  statements  seem  to  be  too  loosely  drawn  to  suit  the 
fastidious  mathematician  my  excuse  must  be  that  I  am  making  no  attempt  at 
a  rigorous  presentation  of  the  subject,  but  am  merely  trying  to  furnish  the 
novice  with  a  useful  instrument  for  a  special  purpose,  one  which  he  can  apply 
at  once,  and  which  is  capable  of  yielding  immediate  results. 

ELEMENTARY  PRINCIPLES. 

A  determinant  is  conventionally  written  in  the  form  of  a  square,  consisting 
of  an  equal  number  of  rows  and  columns  of  figures,  set  between  two  vertical 
lines.  The  number  of  rows  and  columns  may  be  anything  we  please,  but 
for  most  of  our  purposes  a  determinant  of  the  third  order  consisting  of  three 
rows  and  three  columns  will  be  all  that  we  shall  need.  Where  more  than 
three  variables  are  to  be  dealt  with  it  is  frequently  possible,  as  will  be  shown 
later,  to  so  manipulate  them  as  to  use  the  third  order  determinant.  My 
discussion  will  therefore  be  confined  to  the  case  where  the  determinant  con- 
tains but  three  rows  and  three  columns,  and  any  statements  that  I  may  make 
about  the  properties  of  determinants  must  be  understood  to  refer  to  deter- 
minants of  this  order  only. 

no 


THE    USE    OF    DETERMINANTS 


III 


Let 


a  b  c 
d  e  f 
g  h  i 


be  a  determinant  involving  the  quantities  a,  6,  c,  d,  e,  f,  g,  h  and  i. 

To  evaluate  it  multiply  the  quantities  lying  along  each  diagonal  and 
combine  the  products  by  prefixing  the  plus  or  minus  sign  according  to  the 
direction  of  the  diagonals,  those  sloping  down  to  the  right  being  considered 
plus  and  those  sloping  down  to  the  left  minus.  Thus  the  expansion  of  the 
above  determinant  will  read  aei  +  bfg  +  cdh  —  afh  —  bdi  —  ceg.  This  may 
be  more  apparent  if  the  upper  two  rows  are  repeated  below, 
and  the  diagonals  indicated  with  their  proper  signs. 

By  multiplying  along  the  lines  indicated  and  using  the 
attached  signs  the  expansion  will  be  found  to  be  that  in- 
dicated  above. 

The  individual  letters  in  this  case  have  been  supposed  to  be  all  affected 
by  the  positive  sign.  Any  of  them  may  be  negative,  however,  and  the  sign 
of  the  product  in  which  it  appears  will  be  governed  by  the  ordinary  rules  of 
algebraic  multiplication.  Also  any  of  them  may  be  zero,  in  which  case  the 
product  containing  it  will  vanish. 

To  illustrate  these  points  let  us  expand  the  following  determinant. 


3 
6 

-5 


—  2 


=  3X4X1  +  oX(-2)X(-5)+i  X6X3~3X(-2)X3 
-oX6Xi-iX4X(-  5)  =12+0  +  18  +  18-  o 
+  20  =  68 


In  our  work  the  determinant  will  always  be  placed  equal  to  zero.  Subject 
to  this  restriction  the  following  rules  for  the  transformation  of  determinants 
are  given  without  proof. 

1.  A  column  may  be  interchanged  with  another  column,  or  a  row  with 
another  row,  without  altering  the  value  of  the  determinant. 

2.  A  column  may  be  added  to  another  column,  or  a  row  to  another  row, 
without  altering  the  value  of  the  determinant. 

3.  A  column  or  row  may  be  multiplied  by  any  quantity  without  altering 
the  value  of  the  determinant. 

To  illustrate,  take  the  following  determinant  which  has  been  made  equal 
to  zero. 


6  i       o 

7  2   -5 
9    2   -3 


6  X  2  X  (-3)  +  i  X  (-5) 
-iX7X(-3)-oX 

21   —  O  =  O 


oX7X2-6X(-5)X2 


112 


CONSTRUCTION    OF    GRAPHICAL    CHARTS 


First,  interchange  the  second  and  third  columns. 


o       o  i 

7   -5  2 
9  ~3  2 


=  6  X  (-5)  X2+oX2X9+iX7X  (-3)  -  6  X  2  X  (-3) 
-oX7X2-iX  (-5)  X  9  =  -60  +  0-21+36-0 
+  45  =  ° 


Then  add  the  first  column  of  the  first  determinant  to  the  second  column. 


670 
7  9  ~5 
9  ii  -3 


=  6  X  9  X  (-3)  +  7  X  (-5)  X9+°X7Xn- 
6  X  (-5)  X  ii  -  7  X  7  X  (-3)  -0X9X9  = 
-  162  -315  +  o  +  330  +  147  -  o  =  o 


Finally,  multiply  the  third  column  of  the  first  determinant  by   —2. 


1  o 

2  10 
2     6 


=  6X2X6  +  1X10X9  +  0X7X2-6X10X2- 
1X7  X6  —  0X2X9  =  72 +  90  +  0—  120—  42  —  0  =  0 


In  each  of  the  above  transformations,  made  according  to  the  rules, 
we  see  that  the  value  of  the  determinant,  which  was  zero,  remains  unchanged. 

Analytic  geometry  tells  us  that  the  three  points  x\y\t  X2y2,  Xsys,  He  upon 
the  same  straight  line  if  they  are  connected  by  the  equation 

xi  —  x2  =  yi  —  y2 
xz  —  #3      yz  —  ys 
Clearing  of  fractions  and  transposing,  this  becomes 

xiyz  —  xiy3  —  x2y2  +  x2yz  =  x2yi  —  x2y2  — 
or 


That  this  same  condition  is  represented  by  the  following  determinant 
is  shown  by  expanding  it. 


x3     y3     i 


(i) 


The  condition  that  three  specified  points  shall  lie  upon  the  same  straight 
line  is  evidently  that  with  which  we  have  to  do  in  the  case  of  the  alinement 
chart.  In  this  chart  there  are  three  strings  of  points  or  three  graduated 
axes  which  must  be  so  arranged  that  a  straight  line  drawn  across  them 
anywhere  will  pass  through  points  on  the  axes  whose  values  will  satisfy  the 
equation  for  which  the  chart  was  constructed. 

The  first  step  in  the  process  is  to  write  the  equation  we  propose  to  chart  in 
its  determinant  form,  that  is  we  write  the  determinant  in  such  a  way  that 
when  it  is  expanded  it  will  give  our  original  equation.  Assuming  that  the 
equation  contains  three  variables,  the  determinant  must  be  written  so  that 
one  variable,  and  only  one,  appears  in  each  row.  The  determinant  is  then 


THE    USE    OF    DETERMINANTS 


modified  in  accordance  with  the  rules  given  above  until  one  of  the  columns  is 
filled  with  ones.  The  other  two  columns  will  then  contain  the  x  and  y  coor- 
dinates of  the  three  strings  of  points  corresponding  to  the  three  variables. 
While  it  is  possible  to  indicate  in  a  general  way  the  methods  to  be  followed 
in  preparing  an  equation  for  charting,  it  will  be  found  that  many  of  them 
present  distinct  peculiarities  of  their  own  which  make  special  problems  of 
them.  The  successful  solution  of  such  cases  will  depend  largely  upon  the 
skill  and  experience  of  the  chart  maker.  A  knack  of  juggling  the  equations 
into  practicable  forms  will  prove  to  be  a  valuable  asset.  In  what  follows, 
the  methods  which  I  have  found  to  be  most  useful  will  be  illustrated  by  a 
series  of  examples,  so  chosen  as  to  bring  out  the  more  important  chart  forms 
and  some  others  of  special  interest,  and  to  illustrate  a  few  of  the  expedients 
which  are  useful  in  handling  refractory  equations. 


A  "Z"  OR  "N"  TYPE  CHART  FOR  SECTION  MODULUS. 

Let  the  first  problem  be  to  chart  the  equation  for  the  section  modulus 
for  rectangular  areas 

_  _  BH* 
6 

where  Z  is  the  section  modulus,  B  the  breadth  and  H  the  depth  of  the  area. 

It  is  generally  best  (though  unnecessary  in  this  case)  to  begin  by  clearing 
of  fractions,  and  then  place  the  equation  equal  to  zero.  The  equation  then 
becomes  6Z  =  BH2  or  6Z  —  BH2  =  o.  It  is  then  put  into  the  determinant 
form,  that  is  we  write  a  determinant  which,  when  expanded,  will  give  our 
equation.  In  doing  this  each  variable  must  occupy  a  row  to  itself,  Z,  say, 
being  in  the  first,  B  in  the  second,  and  H  in  the  third,  no  variable  appearing 
outside  of  its  own  row.  This  determinant  may  be  written  by  inspection  or 
by  resolving  the  original  equation  into  three  equations,  one  for  each  variable. 

To  write  the  determinant  by  inspection  begin  by  putting  6Z  in  the  upper 
left  hand  corner  and  fill  out  the  diagonal  by  a  i  in  the  second  place  in  the 
second  row,  and  i  in  the  third  place  in  the  third  row.  The  product  will 
then  be  6Z.  In  the  second  row  and  directly  below  Z,  put  B,  and  in  the  second 
place  in  the  third  row  H2,  (or  B  and  H*  may  be  interchanged).  Then  if 
we  place  a  minus  one  in  the  third  place  of  the  first  row  we  will  get  the 
product  —  BH2.  The  remaining  places  are  now  to  be  filled  with  zeros, 
the  result  being  shown  below,  together  with  a  third  variation. 

6Z     -i 


6Z  o 

B  i 

o  H 
8 


—  i 
o 

i 


or 


6Z 
H2 
o 


0  —i 

1  O 

B         i 


=   o     or 


o 
H2 


CONSTRUCTION   OF   GRAPHICAL   CHARTS 


It  sometimes  happens,  however,  that  this  simple  and  obvious  method  is 
for  some  reason  unsatisfactory  or  unworkable,  in  which  case  we  try  to  break 
up  the  original  equation  into  three,  one  for  each  variable,  and  make  each 
equation  equal  to  zero.  In  the  present  case  with  the  original  equation 
6Z  -  BH*  =  o,  let  x  =  Z,  or  x  -  Z  =  o;  let  y  =  H  2,  or  y  -  H*  =  o.  Then 
substituting  in  the  original  equation  we  have  6x  —  By  =  o.  This  gives  us 
three  equations  in  x  and  y,  one  containing  Z,  another  H  and  the  third  B. 
These  three  equations  have  the  general  form  ax  +  by  +  c  =  o,  where  a  is 
the  coefficient  of  x,  b  is  the  coefficient  of  y  and  c  is  a  constant,  a,  b  and  c 
may  have  any  value,  positive  or  negative,  or  may  be  zero. 

Lay  out  the  determinant  as  shown  below,  heading  one  column  (a), 
another  (b)  and  the  third  (c).  It  is  immaterial  which  column  is  assigned  to 
which  letter,  because,  as  shown  in  the  rules  for  transformation,  the  columns 
may  be  freely  interchanged. 


(a) 

* 
* 
* 


(c) 


In  the  first  row  insert  in  the  (a)  ,  (b)  and  (c)  columns  the  values  of  these 
quantities  taken  from  the  equation  x  —  Z  =  o.  Here  a,  the  coefficient  of  x,  is 
i,  b,  the  coefficient  of  y,  is  o  and  c  is  —  Z.  In  the  second  row  do  the  same  for 
the  equation  y  —  H2  =  o.  Here  a  =  o,  b  is  i  and  c  is  —  H*.  In  the  third 
row  do  the  same  for  the  equation  6x—By  =  o.  Here  a  is  6,  b  is—  B  and  c  is  o. 

The  resulting  determinant  then  is 


(a) 

I 

O 

6 


o 

i 

-B 


(c) 
-Z 
-H* 

o 


=  o 


(a) 


If  this  determinant  is  expanded  it  will  be  seen  that  it  is  the  equivalent 
of  the  original  equation. 

Referring  back  to  determinant  (i)  it  is  seen  that  we  must  have  a  column 
filled  with  ones,  and  that  the  other  two  contain  the  x  and  y  coordinates  of  the 
points  on  our  graduated  axes.  The  determinant  just  written  lacks  a  column 
of  ones,  but  this  may  be  remedied  by  a  little  manipulation.  //  should  be 
noted  here  that  after  the  determinant  is  once  formed  the  (a) ,  (b)  and  (c)  designa- 
tions of  the  columns  may  be  disregarded  as  having  no  special  meaning  and  that 
we  may  assign  any  column  we  please  to  x}  any  other  to  y  and  the  third  to  the  ones. 
Suppose  that  we  decide  to  make  the  first  column  in  (2)  the  "one"  column. 
It  contains  a  zero  in  the  second  row  which  we  must  get  rid  of.  This  is  done 


THE    USE    OF    DETERMINANTS 


by  the  addition  or  subtraction  of  one  of  the  other  columns  which,  according 
to  rule  (2)  will  not  affect  the  value  of  the  determinant. 

It  will  make  considerable  difference  which  column  is  used  and  whether 
we  add  or  subtract.  Experience  will  be  the  best  guide  in  this,  and  as  a  full 
discussion  of  the  different  effects  would  take  up  more  room  than  can  be 
spared  here  I  shall  confine  myself  to  the  explanation  of  such  points  as  come 
up  in  the  development  of  the  problems  which  follow.  In  this  case  let  us 
subtract  the  quantities  in  the  second  column  of  determinant  (2)  from  the 
quantities  in  the  same  rows  in  the  first  column.  The  result  of  this  subtrac- 
tion is 

i  o      — Z 

-i  i      -H*    =o 

6+B     -B         o 

Now  divide  each  row  by  the  quantity  in  that  row  in  the  first  column, 
(see  rule  3)  that  is,  divide  the  first  row  through  by  i,  the  second  by  —  i  and 
the  third  by  6+B.  Then  (as  a  matter  of  convenience,  only)  multiply  the 
second  column  by  —  i .  These  steps  are  shown  below. 

(*)      (y) 


I 

o 

-Z 

i 

o 

-Z 

I 

—  I 

H2 

— 

i 

i 

H2 

-B 

B 

6  +  1 

6  +  B 

We  have  thus  obtained  a  determinant  with  the  necessary  column  of  ones 
and  must  consider  the  other  two  columns  to  contain  the  x  and  y  coordinates 
of  the  points  on  our  graduated  axes.  It  is  of  no  importance  which  column 
is  assigned  to  x  and  which  to  y,  but  as  it  is, usually  desired  to  have  the  principal 
scales  vertical  (here  the  scales  for  Z  and  H)  we  will  call  the  third  the  y  column 
and  the  second  the  x,  as  indicated  by  the  letters  at  the  tops  of  the  columns. 

The  coordinates  for  the  three  sets  of  points  are  to  be  taken  direct  from 
the  determinant  and  are 

B 


y=-Z 


(H) 


y  = 


6  +  B 

y  =  o 


These  equations  are  to  be  interpreted  by  the  usual  methods  of  analytic 
geometry,  and  mean  that,  since  x  is  zero  for  all  values  of  Z,  Z  is  to  be  plotted 
on  the  Y-axis  and  downward  on  account  of  the  negative  sign;  H  is  to  be  plotted 
on  a  vertical  line  at  a  distance  of  one  unit  to  the  right  of  the  origin  and  upward 
on  account  of  the  positive  sign  of  H  in  the  y  equation.  The  y  equation  also 
shows  that  the  H  scale  is  to  be  laid  off  in  terms  of  the  square  of  H .  Since 
y  for  B  is  zero  it  is  clear  that  B  must  be  plotted  on  the  X-axis,  and  since 


Unit  > 


Il6  CONSTRUCTION   OF   GRAPHICAL    CHARTS 

x  for  B  is  always  less  than  one,  and  positive,  the  B  graduations  will  all  lie 
between  the  Z-  and  H-axes.  The  general  scheme  for  the  three  scales  will 
therefore  be  that  shown  in  the  sketch  in  Fig.  59. 

The  next  point  demanding  our  attention  is  the  size  of  the  chart  and  the 
limiting  values  of  the  variables.     It  is  never  desirable  to  make  a  chart  more 
comprehensive  than  is  demanded  by  the  conditions  under  which  it  is  to  be 
used.     We  usually  have  certain  definite  limits  set  for  the 
size  of  the  chart,  and  if  the  limiting  values  of  the  variables 
are  set  farther  apart  than  is  really  necessary  there  will 
be  a  sacrifice  of  accuracy  in  reading  on  account  of  the 
x      smaller   sized   divisions    used   in   representing   a   given 
value. 

As  regards  the  charts  shown  here  as  illustrations  of 
methods  we  are  not  hampered  by  any  consideration  of 
IFIG.  59.  possible  practical  service  for  them.     The  only  require- 

ment is  that  they  shall  show  clearly  the  points  under 
discussion,  and,  subject  to  this,  the  choice  of  limiting  values  will  be  purely 
arbitrary. 

In  the  present  problem  let  us  assume  that  B  is  to  vary  from  o  to  8,  and 
H  from  o  to  12.  Then  Z  will  vary  from  o  to  192.  Suppose  that  in  this 
chart  and  those  which  follow  we  limit  the  length  of  our  vertical  scales  to  about 
3".  The  y  equation  for  Z,  as  given  above,  is  y  =  —  Z.  Here  y  represents  a 
measured  length  in  inches  whose  maximum  value  must  be  192"  if  the  equa- 
tion is  left  as  it  is.  To  make  this  come  right  let  us  alter  the  equation  to  read 

_  £ 
y  =  — —  which  means  that  each  value  of  Z  we  wish  to  plot  must  first  be 

divided  by  60  to  get  the  scaled  length  in  inches.  Or  we  may  lay  the  60  side 
of  the  engineer's  scale  along  the  axis  and  graduate  directly  from  that.  It  is 
obvious  that  the  latter  is  the  logical  thing  to  do,  since  it  saves  all  calculation, 
and  it  is  equally  obvious  that  in  order  to  do  this  we  must  choose  for  the 
denominator  of  our  fraction  some  figure  which  is  found  on  the  engineer's  or 
other  scales  at  our  command.  To  avoid  future  misunderstanding,  attention 
should  be  called  to  the  fact  that  the  scale  unit  used  here  is  not  1/60"  but  i", 
since  on  substituting  for  Z  in  the  equation  y  is  expressed  in  inches.  This 
same  i"  scale  unit  must  be  used  for  all  vertical  scales. 

In  a  like  manner,  with  a  maximum  value  of  12  for  H,  H2  =  144,  and 

H2 
we  shall  keep  to  our  limiting  3"  length  by  writing  the  equation  y  =  — . 

The  squares  of  H  are  therefore  to  be  laid  off  from  the  50  side  of  the  engineer's 
scale.  The  real  scale  unit  is,  however,  i"  as  before.  I  am  laying  special 


THE    USE    OF    DETERMINANTS 


117 


emphasis  on  this  point  because  I  find  that  beginners  are  very  likely  to  become 
confused  with  the  apparent  changes  in  scale  unit  for  different  parts  of  the 
chart,  and  do  not  know  what  to  do  when  there  is  no  denominator  to  guide 
them. 

To  make  these  changes  we  must  go  back  to  the  original  determinant  (3). 
We  are  privileged  to  multiply  or  divide  columns  as  we  please  (rule  3),  and  it  is 
apparent  that  we  could  get  the  desired  result  so  far  as  Z  is  concerned 
by  dividing  the  third  column  by  60.  But  this  would  also  divide  H2  by 
60,  whereas  we  want  to  divide  it  by  50.  A  little  reflection  will  show  that 
we  will  get  the  same  effect,  so  far  as  Z  is  concerned,  by  multiplying  the  first 
column  of  the  determinant  (2)  by  60  and,  since  the  second  row  contains 
zero  in  that  column,  H  will  not  be  affected.  Likewise  if  we  multiply  the 
second  column  by  50  we  shall  get  the  desired  result  as  regards  H  without 
affecting  Z.  We  will  therefore  make  these  alterations  in  (2)  and  then  proceed 
exactly  as  before,  the  steps  being  indicated  below. 


-Z 


60         o       -Z 
o       50      -H2 
360    —  $oB        o 

= 

60                o 
-50               50 

360+505  -SOB 

= 

1. 

z 
I 

-Z 

60 

H2 

5° 

360+50.6      ° 

I 

X 
I 

(*) 

0 

I 
5* 

60 
-z 

60 
m 

5° 
o 

36+5^ 

The  coordinates  taken  from  the  last  determinant  now  read 


(Z) 


(H) 


X  =   I 


H 


In  plotting  this  chart  it  should  be  noted  that  there  is  no  necessary  con- 
nection between  the  size  of  the  scale  units  used  on  the  horizontal  and  vertical 
axes.  The  vertical  scale  unit  has  been  decided  upon  as  i",  but  the  scale 
unit  for  plotting  B  along  the  X-axis  may  be  anything  else  we  wish.  Our 
equations  show  that  the  distance  between  the  Z  and  H  scales  is  to  be  one  unit, 


Il8  CONSTRUCTION   OF   GRAPHICAL   CHARTS 

and  since  we  are  free  to  choose  its  size  we  will  make  it  whatever  will  give  a 
good-looking  chart.  For  reasons  which  will  appear  later  let  us  make  the 
length  of  the  unit  for  the  x  measurements  3^"-  This  length  is  represented 
by  loo  on  the  30  scale,  which  will  be  convenient  for  laying  off  the  decimal 
fractions  which  result  from  the  solution  of  the  B  equation. 

The  chart  now  looks  something  like  (a)  of  Fig.  60,  and  an  index  line 
drawn  so  as  to  cut  the  three  scales  will  do  so  at  points  which  satisfy  the 
equation.  But  it  will  be  seen  that  such  a  chart  occupies  a  good  deal  of 
space.  This  may  be  reduced  by  the  following  simple  expedient.  Suppose 
the  Z  and  H  scales  to  be  hinged  at  the  points  where  they  join  X,  and  that  the 
Z  scale  is  shoved  up  as  shown  at  (b)  Fig.  60,  carrying  with  it  the  B  scale  on  the 

X-axis.  No  change  has  been  made  in  the 
length  or  graduation  of  any  of  the  scales. 
This  being  so  any  three  points  which  lie  on 
an  index  line  in  the  first  figure  will  be  in 
alinement  in  the  second.  It  is  at  once 
apparent  that  a  great  improvement  has 
been  made  in  compactness  and  that  we 
have  obtained  a  form  of  the  Z,  or,  as  it  is 
sometimes  called,  the  N  type  of  chart  dis- 
FIG.  60.  cussed  in  Chapter  V.  If  the  distance  be- 

tween Z  and  H,  measured  along  X,  is  made 

a  little  greater  than  the  lengths  of  the  vertical  scales  it  should  be  possible 
to  place  the  latter  nearly  opposite  to  each  other.  The  reason  for  choosing 
3  1/3"  for  this  length  should  now  be  apparent.  The  actual  angle  of  inclina- 
tion for  the  X-axis  is  not  of  great  importance.  45°  would  probably  be  best  if 
there  is  room  to  make  the  chart  square,  but  as  we  usually  want  the  chart  to 
be  longer  than  it  is  wide,  especially  where  it  is  to  fit  a  printed  page,  the  angle 
will  be  greater. 

The  scales  are  now  ready  for  graduation,  and  this  is  a  very  simple  matter. 
The  60  side  of  the  engineer's  scale  is  laid  along  Z  and  the  desired  divisions  are 
picked  off  and  marked.  H  is  laid  off  from  the  50  scale  in  terms  of  the  square 

ZT2  Q-p 

of  H.     For  instance  with  H  =  9,  y  =  --  =  — .     81  will  therefore  be  marked 

from  the  50  scale,  but  will  be  lettered  9  and  not  81.  The  graduation  of  B 
will  be  slightly  more  troublesome.  It  requires  the  solution  of  the  equation 

x  =     ,   .     p  for  each  value  of  B  required.     For  instance  for  B  =  2,  #  =  0.217; 

for  B  =  4,  x  =  0.357,  etc.  These  values  are  to  be  laid  off  on  the  X-axis, 
the  unit  length  being,  as  explained  before,  100  on  the  30  side  of  the  engineer's 


THE   USE    OF   DETERMINANTS 


12 


Pole 


scale,  and  the  zero  at  the  left  hand  end  of  the  scale  where  it  meets  Z.     The 

finished  chart  is  shown  in  Fig.  61. 
If  the  whole  number  values 

of  B  between  o  and  8  were 

the  only  ones  to  be  laid  off  it 

would  be  no  great  hardship  to 

calculate  and  lay  them  off  as 

just     described,    but    if    the 

number  of  graduations  is  large 

this   becomes  something  of  a 

chore.     Fortunately  there  is  a 

very  simple  method  of  avoiding 

this    labor.     An   equation   of 

the    general    form    of    the   x 

equation    for    B    will   give  a 

scale  which  is  projective.     For 

instance  if  we  have  two  inter- 
secting lines,  as  in  Fig.  62  one 

of  which  carries  the  "regular" 

or  usual  graduations  of  some 

function,   and   if   from    these 

graduations  we  draw  lines  to 

some   pole,    the   intersections 

of  these  lines  with  the  second 

line  will  give  us  a  projected 

scale  of  the  function  on  this  line.     Necessarily  these  two  scales  must  intersect 

at  the  point  having  the  same  number  on  each. 

In  the  present  case  the  Z  and  B  scales 
intersect  at  a  point  which  is  zero  for  each, 
and  they  are  functions  of  the  same  char- 
acter, hence  it  should  be  possible  to  project 
B  from  Z.  The  size  of  the  unit  on  Z  from 
which  to  make  our  projections  is  not 
important.  In  this  case  let  us  call  the 
length  of  the  unit  20  on  Z.  The  eight 
divisions  required  on  B  will  then  carry  us 
up  to  1 60  on  Z.  Now  find  two  or  three 
points  on  B  by  calculation,  say  4,  6,  and  8, 

for  which  x  will  be  0.357,  0.455  and  0.526.     Join  them  to  the  points  80,  120 

and  1 60  on  Z  (corresponding  to  4,  6,  and  8)  and  if  the  points  have  all  been  cor- 


FIG.  61. — Chart  for  section  modulus.' 


I2O 


CONSTRUCTION    OF    GRAPHICAL    CHARTS 


rectly  located  it  will  be  found  that  the  lines  joining  them  will  intersect  at  a 
point,  the  point  lying  on  the  H-axis.  This  point  may  now  be  taken  as  the  pole 
for  projecting  as  many  other  points  from  Z  to  B  as  are  wanted.  If  we  wanted 
quarter  inch  graduations  on  B  instead  of  whole  number  divisions  we  could 
project  them  from  the  five  marks  on  Z  instead  of  the  twenties. 

It  is  sometimes  desirable  to  run  the  graduations  on  the  intermediate 
scale  beyond  the  limits  originally  set  for  it.  B,  for  instance,  might  easily 
be  extended  beyond  the  original  value  of  8  without  taking  up  any  additional 
room  on  the  chart  space.  These  higher  values  of  B  could  not  be  used  in 
connection  with  the  maximum  values  of  H  as  the  Z  readings  would  fall 
outside  the  limits  of  the  paper,  but  it  is  sometimes  very  convenient  to  have 
them  for  use  with  the  lower  values  of  H.  To  do  this  it  will  be  necessary 
to  project  from  another  pole.  In  this  instance  we  will  use  a  unit  length  of  10 
on  Z  to  project  from.  Draw  lines,  therefore,  through  40,  60  and  80  on  Z  and 
4,  6  and  8  on  B.  They  will  meet  at  another  pole  (again  on  H)  from  which  we 
can  project  B  graduations  up  to  20  if  we  wish  to  do  so.  Fig.  61  shows  the 
projecting  lines  for  15  and  20,  using  the  new  pole. 

The  poles  on  H  may  be  found  by  calculation  as  well  as  by  trial.  For 
the  first  case  let  Z  =  2oB  and  substitute  this  for  Z  in  the  equation  Z  = 

7?TT2 


Then  2oB  =  —r-  or  H2  =  120,  and  H  =  10.95.     The  pole  will  then 

be  located  120  divisions  on  the  50  scale  from  the  H  zero,  or  at  the  point 
corresponding  to  10.95.  I*1  the  second  case  let  Z  =  loB.  Then  loB  = 

7JTT2 

—,--  or  H2  =  60,  and  H  =  7.75.     The  points  thus  found  should  correspond 

exactly  with  the  poles  which  were  located  graphically  if  the  work  has  been 
carefully  done.  It  is  not  a  bad  plan  to  use  both  methods  as  a  check  on  the 
work. 

A  MODIFICATION  OF  THE  PREVIOUS  CHART. 

Let  us  now  return  to  the  original  determinant  (2)  and  see  what  effects 
will  result  from  a  different  treatment  of  it.  Suppose  that  we  add  the  second 
column  to  the  first  instead  of  subtracting.  Then  divide  each  row  by  the 
quantity  in  the  first  column  and  multiply  the  third  by  —  i. 

(*)    (y) 


i       o   -Z 
o       i  -H2 
6  -B      o 

= 

i           o  -Z 
i           i  -H2 
6-B  -B       o 

i      o      —  Z 

i      i     -H2 
—  B 

= 

i      o       Z 

i      i      H2 
-B 

1  6-B      ° 

1  6-B     ° 

THE    USE    OF   DETERMINANTS 


121 


From  this  we  write  the  construction  equations 


(Z) 


=  o 


(H) 


X  =   I 


-B 
6-B 

y  =  o 


These  equations  show  that  Z  and  H  are  to  be  plotted  as  before  on  vertical 
axes  one  unit  apart,  but  in  this  case  the  positive  signs  in  the  y  equations  show 
that  both  scales  are  to  lie  above  the  X-axis.  B  again  lies  on  X,  but  instead 
of  being  placed  between  Z  and  H  it  will  be  exterior  to  them.  If  B  is  less 
than  6,  x  will  be  negative  and  laid  off  to  the  left  of  Z.  If  B  =  6,  x  will  be 
infinity.  And  if  B  is  greater  than  6,  x  will  be  positive;  it  will  also  be  greater 
than  one  and  hence  will  lie  to  the  right  of  H. 

Since  the  values  of  B  we  wish  to  chart  are  both  larger  and  smaller  than  6 
such  a  scale  is  clearly  impracticable,  nevertheless  it  may  be  put  into  a  practic- 
able form  by  a  little  manipulation  of  the  determinant.  If,  instead  of  6 


in  the  denominator  of  the  fraction 


-B 


we  had  a  number  greater  than  8 


6  -B' 

(the  assumed  maximum  for  B)  the  scale  would  lie  wholly  to  the  left  of  Z, 
while  if  the  6  were  changed  to  something  less  than  i  (the  assumed  minimum 
for  B)  our  scale  would  lie  wholly  to  the  right  of  H . 

For  instance  let  us  multiply  the  first  column  of  determinant  (2)  by  2. 
Then  add  the  second  column  to  the  first,  divide  by  the  first  column,  and 
then,  to  get  the  H  scale  within  the  previously  specified  3"  length,  divide  the 
third  column  by  —50.  These  steps  are  indicated  below. 


o   -  Z 


i   - 


12  -B 


o   -Z 


-H2  = 


I2-B  -B 


-Z 

(*)     (y) 
z 

= 

I          O 

2 

i       i         -H2 
-B 

— 

I           O 
100 

H2 

i       i 
5° 
-B 

1  I2-B 

1   I2-B    ° 

=  0 


From  this  the  construction  equations  are  written 


(Z) 


(H) 


H1 


(B) 


x  = 


-B 


I2-B 

y  =o 


According  to  these  equations,  Z  is  to  be  laid  off  from  the  10  scale  (really 
i"  =  100),  H  from  the  50  scale,  and  B  in  an  arbitrary  unit  whose  length  is 
the  distance  assumed  between  Z  and  H  measured  along  the  X-axis.  The 
X-axis  will  be  sloped  again  for  the  sake  of  a  better  appearance.  B  may  be 


122 


CONSTRUCTION    OF    GRAPHICAL   CHARTS 


laid  off  by  calculation,  in  which  case  its  maximum  and  minimum  values  are 


found  to  be 


-  8 


—  2  and 


—  i 


—  0.0909,  or  it  may  be  projected  as 


12—8  12  —  I 

before  from  a  pole  found  graphically  or  by  calculation.     The  result  is  shown 
in  Fig.  63. 

As  a  matter  of  interest  we  might  try  to  see  what  will  happen  if  other 

12T12 


FIG.  63. — Chart  for  section  modulus. 

columns  are  added  together.     For  instance  in  determinant  (2)  add  the  second 
column  to  the  third  and  divide  by  the  third. 

60     (*) 

—  I 

~Z 


i       o   -Z 
o       i   -H* 
6  -B       o 


i  o  -Z 
o  i  i-H 
6  -B  -B 


I  -if' 


-6 

-IB 


The  construction  equations  from  this  are 


(Z) 


X  =  O 

—  I 


(H) 


(B) 


-6 


A  very  brief  inspection  will  show  that  a  chart  constructed  in  accordance 
with  these  equations  will  have  some  most  undesirable  features.     All  of  the 


THE   USE    OF   DETERMINANTS 


I23 


variables  will  be  represented  by  reciprocal  scales,  a  thing  which  should  always 
be  avoided  if  the  limiting  values  of  the  variables  are  much  separated,  as  it 
leads  to  an  undesirable  crowding  of  the  graduations  for  the  higher  values. 
This  alone  would  be  enough  to  condemn  this  chart  without  further  investiga- 
tion. A  considerable  number  of  other  combinations  can  be  made  which  are 
not  discussed  here  from  a  lack  of  space,  but  with  which  the  reader  may 
profitably  experiment  and  in  this  way  accumulate  a  valuable  fund  of  experi- 
ence which  will  be  useful  for  future  work. 

!  A  CHART  FOR  THE  FRANCIS  WEIR  FORMULA. 

The  next  chart  will  be  one  having  a  curved  scale,  the  equation  used  being 
the  Francis  weir  formula. 

Q  =  3-33(£  -  c 
This  may  be  rewritten 

Q  -  3-33LH*  + 

Let  x  =  Q  or  x  —  Q  —  o,  and  y  =  Lory  —  L  =  o.  Then  substituting  we 
have  x  —  3.33^^  -ro.666H^  =  o,  and  the  determinant  will  be,  using  the 
same  notation  as  before, 

(a)        (b)  (c) 

i          o  -Q 

0  i  —L         =  o 

1  -3.33^0.666^ 

Suppose  that  we  desire  to  have  the  H  scale  between  the  scales  for  L  and 
Q.  This  may  be  accomplished  by  subtracting  the  second  column  from  the 
first.  Let  us  do  this,  divide  by  the  first  column  and  multiply  the  second  by 
—  i. 

0  —  Q  i  o  —  Q 

1  —  L     =   1—1  L 


(i) 

o 
3-33*" 


w 

-Q 

L 

0.666J794 


=  o 


Then  assigning  x  and  y  to  the  columns  indicated  we  may  write  the  con- 
struction equations  as  follows: 


(0 


x  =  o 


(L) 


=  -Q 


X  =   I 


y=L 


(ff) 


124 


CONSTRUCTION   OF   GRAPHICAL   CHARTS 


They  show  that  Q  is  plotted  downward  on  the  Y-axis;  that  L  is  plotted 
upward  on  a  vertical,  one  unit  to  the  right  of  F;  while  H  is  plotted  on  a  curved 
line  which  is  roughly  parallel  to  X. 

Let  us  take  as  the  limits  for  the  variables  L  —  o  to  6,  H  =  0.5  to  2,  which 
will  give  Q  =  o  to  about  50.  To  bring  the  verticals  within  the  previously 

assumed  length  of  3"  the  y  equations  should  read  y  =  —  —  and  y  =  —     We 

can  get  these  results  by  multiplying  the  first  column  of  the  original  deter- 
minant by  20  and  the  second  by  2  and  then  proceeding  as  before. 


20  O 

O  2 

2O      — 


-e 

-L 


20 
—2 


-Q 

-L 


(*) 

(y) 

I 

0 

nQ 

20 

L 

I 

i 

2 

6.66HK 

0.666H& 

20+6.  66#%      2o+6.66#3/i 

With  these  changes  the  construction  equations  now  are 


(Q) 


x=  o 


*-=» 

20 


(L) 


X  =    I 

L 

y  =  - 


(H) 


6.66H* 


'       20+6.66^ 


The  resulting  chart  is  shown  in  Fig.  64.  Q  has  been  laid  downward  on  the 
Y-axis  from  the  20  side  of  the  engineer's  scale,  and  one  unit  to  the  right,  meas- 
ured along  the  sloping  X-axis,  a  second  vertical  has  been  drawn  upward  for 
L,  the  20  scale  being  used  again,  but  with  the  understanding  that  it  is  half 
an  inch  and  not  one-twentieth  which  represents  a  unit  value  for  L. 

As  in  the  previous  problem  we  will  make  the  unit  length  for  X  &i"  as 
this  gives  a  convenient  width  for  the  chart  and  is  a  length  which  can  be  easily 
divided  into  hundredths  by  the  use  of  the  30  scale. 

We  must  first  graduate  X  in  accordance  with  the  equation  x  =        ' 


6 

Solve  this  for  several  values  of  H,  say  i,  1.5  and  2,  and  we  shall  have  for 
x  0.25,  0.379  and  0.485.  These  are  to  be  laid  off  from  the  left  hand  end  of 
X  using  the  30  scale.  The  rest  of  the  H  graduations  had  better  be  laid  off 
by  projection,  and  since  H  in  the  equation  appears  as  a  three^halves  power 
we  must  project  from  a  three-halves  power  scale.  We  have  nothing  of  the 


THE    USE    OF    DETERMINANTS 


125 


sort  ready  made  and  so  must  lay  one  off.  This  is  shown  in  Fig.  64.  It 
intersects  X  at  the  origin  and  runs  in  any  direction.  On  this,  in  any  con- 
venient scale,  lay  off  the  three-halves  powers  of  the  desired  values  of  H 
from  0.5  to  2.  A  table  of  three-halves  powers  such  as  is  found  in  Mark's 
Handbook  will  be  a  great  convenience  here  From  the  points  on  this  scale 
corresponding  to  i,  1.5  and  2  draw  lines  through  the  same  points  on  X  and 
extend  them  to  their  intersection.  This  point  is  the  pole  from  which  the  rest 
of  the  points  may  be  projected  to  X. 

From  the  graduations  thus  found  on  X,  vertical  lines  are  now  drawn 

and  the  calculated  values  from  the  equation  y  =  — '  ^id  off  on 


ole 


6-  -5 


4:  -4 


Aux.  — •    Power  Scale          X   \          "  " 


0^-0 
FIG.  64. — Chart  for  Francis  weir  formula. 


2-  -2 


them.  It  is  needless  to  calculate  more  than  three  or  four  of  these  quantities, 
say  for  0.5,  i,  1.5,  and  2.  A  smooth  curve  is  drawn  through  the  points  thus 
found,  the  graduations  on  it  being  marked  where  the  curve  is  cut  by  the 
verticals  from  X.  The  scale  unit  used  for  laying  off  the  y  values  for  H  is 
the  inch;  if  there  is  any  difficulty  in  seeing  why  this  is  so  it  should  be  re- 
membered that  this  is  really  the  unit  used  for  y  when  laying  off  L  and  Q, 
and  that  we  used  the  20  scale  merely  to  save  the  labor  of  dividing  by  2  and 
20  for  each  graduation,  y  in  all  cases  is  expressed  in  inches.  To  read  the 
chart  join  the  selected  values  of  L  and  H  and  prolong  the  line  to  the  Q  scale. 


126 


CONSTRUCTION    OF   GRAPHICAL   CHARTS 


As  a  matter  of  passing  interest  it  may  be  pointed  out  that  the  auxiliary 
scale  for  H  on  X  can  be  used  instead  of  the  curve  to  solve  the  formula  Q  — 
3.33!,^,  which  is  used  where  the  weir  has  no  end  contractions.  This  will 
be  apparent  if  the  equation  is  put  into  the  determinant  form. 


A  HYDRAULIC  CHART  HAVING  ONLY  Two  SCALES. 

Another  hydraulic  chart  of  an  interesting  type  is  shown  in  Fig.  65.  It 
is  for  the  velocity  of  flow  from  a  rectangular  orifice  in  a  vertical  wall,  the 
equation  for  which  is 


FIG.  65.  —  Chart  for  flow  of  water  through  rectangular  orifices. 


where  V  is  the  velocity,  and  HI  and  H2  the  distances  of  the  lower  and  upper 
edges  from  the  surface  of  the  water. 

2  I 

Let  us  call  -\fTg  =  8;  then  the  constant  -  X  8  will  be  -^-^  —  >  and  after 

clearing  of  fractions  the  equation  becomes 

0.18757^1  -  0.18757^2  -  Hp  +  HP  =  o 

Let  us  write  the  determinant  for  this  by  inspection,  taking  care  to  keep 
each  variable  in  a  row  by  itself,  V,  say,  in  the  first,  HI,  in  the  second,  and 


THE    USE    OF    DETERMINANTS 


I27 


in  the  third.     Put  7  in  the  first  diagonal  with  HI  and  the  same  7  with 
H2  in  the  opposite  one. 

0.18757     *       * 
*         #1     i 


Then  if  we  put  i  in  the  second  column  of  the  first  row  and  E-fi  and  R^ 
in  the  first  column  of  the  second  and  third  rows  we  have  only  to  fill  the 
last  column  of  the  first  row  with  a  zero  to  complete  the  figure. 

0.18757     i 


H 


=  o 


We  may  now  get  the  necessary  column  of  ones  by  taking  the  determinant 
as  it  stands  and  dividing  by  the  first  or  second  column,  or  we  may  fill  the 
third  column  by  addition  from  either  the  first  or  second  and  divide  by  it. 
In  this  case  let  us  divide  by  the  second  column. 

(y)          (*) 

0.18757     i      o 


=  o 


From  this  the  construction  equations  are 

I  y  =  0.18757  [y  =  HJ*  (y  =  HJ* 

It  will  be  seen  at  once  that  these  equations  give  rise  to  a  very  peculiar 

condition  since  the  expressions  for  HI  and  H2  are  identical;  this  means  that 

the  scales  for  the  two  quantities  coincide. 

Let  the  limits  for  HI  and  H2  be  25  and  4;  the  limits  for  7  will  then  lie, 

roughly,    between    15  and  40.     With  7  =  40,  y  =  0.18757  =  7.5.     This 

divided  by  3  will  give  a  scale  2^"  long  reckoned  from  the  origin,  and  the 

equation  will  read  y  = —  =  0.06257  =  — •     To  effect  this  change 

divide  the  first  column  of  the  last  determinant  by  3 ;  this  will  of  course  affect 
the  H  quantities  in  the  first  column  in  the  same  manner,  and  their  y  equations 


.... 
will  be  y  = 


, 
and  y 


3  3 

As  before,  any  convenient  unit  may  be  used  for  graduating  the  x  values 

of  H.     When  H  =  4,  x  =  77  will  be  a  maximum,  its  value  being  1/4  or  0.25". 


128  CONSTRUCTION    OF    GRAPHICAL    CHARTS 

If  we  choose  a  unit  length  of  10"  x  will  be  2.5".  As  a  matter  of  convenience 
and  for  good  appearance  let  the  X-axis  be  sloped  at  an  angle  of  30°,  and  on 
it  lay  off  the  various  values  of  H  desired  from  a  table  of  reciprocals,  using  the 
10  scale.  Draw  verticals  at  each  of  these  points  and  on  them  lay  off  the 

corresponding  values  of  H**  from  a  table  of  square  roots,  using  the  30  scale, 

y 
and  join  the  points  thus  found  by  a  smooth  curve.     Since  y  =  —  it  is  ap- 

parent that  the  V  scale  may  be  graduated  in  sixteenths  of  an  inch  from  the 
architect's  scale. 

To  read  the  chart  find  the  desired  values  of  HI  and  Hz  on  the  H 
curve,  join  them  and  prolong  the  line  to  its  intersection  with  V.  The  point 
thus  found  will  be  the  solution  of  the  equation. 

It  is  scarcely  necessary  to  point  out  that  this  chart  has  no  great  practical 
value  except  where  there  is  a  considerable  difference  between  HI  and  H2. 
If  they  are  close  together  the  index  line  will  be  difficult  to  locate  with  any 
degree  of  accuracy,  but  in  this  connection  an  interesting  peculiarity  of  the 
chart  should  be  noted.  If  the  orifice  is  so  narrow  that  HI  and  Hz  are  substan- 
tially the  same  the  equation  fails  since  we  have  zero  over  zero,  which  is 
indeterminate.  The  chart,  however,  will  give  a  solution  if  we  draw  a  tangent 
to  the  curve  at  the  chosen  value  of  H  and  extend  it  to  V. 

A  LOGARITHMIC  CHART  FOR  THE  MODULUS  OF  RESILIENCE. 

Many  equations  which  involve  simple  multiplication  or  division  are 
best  represented  by  charts  having  logarithmic  scales.  As  an  illustration  of 
this  type  take  the  equation  for  the  modulus  of  resilience 


where  M  is  the  modulus,  /  the  fiber  stress  at  the  elastic  limit  and  E  Young's 
modulus  of  elasticity. 

Write  the  equation  2EM  =  /*,  or  log.  2  +  log.  E  -f-  log.  M  =  2  log./;  then 
letting  x  =  log.  E  and  y  =  log.  /  we  will  have  log.  2  +  x  +  log.  M  =  2y.  Put 
these  three  equations  equal  to  zero  and  insert  the  coefficients  of  x  and  y  and 
the  constants  in  their  proper  columns  in  the  determinant  as  explained  before. 

(«)      («         -to 
i        o      —log.  E 

0  I       —log./  =  o 

1  —2          log.  2  +log.  M 

Let  us  suppose  that  we  wish  to  put  the  M  scale  between  those  for  E  and/. 
It  should  be  apparent  that  if  the  second  column  is  subtracted  from  the  first, 


THE    USE    OF    DETERMINANTS 


129 


and  the  rows  divided  by  the  quantities  in  the  first  column  we  shall  have  a 
quantity  in  the  second  column  and  third  row  which  is  less  than  unity;  in 
the  second  row  this  will  be  i  and  in  the  first  o.  If,  then,  the  second  column 
is  taken  for  x,  E  will  be  located  on  the  Y-axis,  /  will  be  at  unit  distance  from 
it,  while  M  is  at  2/3  this  distance. 

These  operations  are  indicated  below. 


—  i 


—  2 


-log.E 
-log./ 

log.  2  +  log. 

M 

I 
I 
I 

(* 

0 

I 

2 

i         o          —  log.  E 

I        -  I                    log.  / 
—  2       log.  2  +  log.  M 

3 

)          GO 

-log.E 

log./ 
log.  2+  log.  M 

3 
=  o 

3                3 

From  this  the  construction  equations  are 


(E) 


x  =  o 


(M) 


y  =  —  log.  E 


y  =  log.  f 


2 

X  =  - 

3 


_  log.  2+ log.  M 
~~ 


These  equations  show  that  £,/and  M  are  all  logarithmic  scales  plotted  on 
vertical  lines.  As  regards  limiting  values  it  is  clear  that  we  cannot  run  down 
to  zero  since  the  logarithm  of  zero  is  minus  infinity.  This  restriction  is  of  no 
importance  here  since  none  of  the  variables  even  approach  zero  in  practice; 
let  us  assume  that  E  varies  from  25  000,000  to  35,000,000  and/ from  10,000  to 
200,000.  Then  M  will  lie  between  1.4  and  800,  or  in  round  numbers  from  i  to 
1000  and  the  logarithm  will  vary  from  o  to  3.  The  logarithm  of  25,000,000  is 
7.398  and  of  35,000,000  is  7.544,  the  difference  being  0.146  which  will  be  the 
length  shown  on  the  scale.  The  logarithm  of  10,000  is  4  and  of  200,000  is 
5.301,  a  difference  of  1.301.  To  represent  1.301  on  our  usual  3"  length  we 

should  divide  by  0.4,  making  the  equation  read  y  =          >  and  this  result  can 

0.4 

be  obtained  by  multiplying  the  second  column  of  the  first  determinant  by  0.4. 
The  length  of  the  E  scale  is  relatively  unimportant,  the  range  in  values  is  so 
small  that  we  do  not  need  a  long  scale  to  represent  it.  Moreover,  as  E  is  in- 
creased in  length  there  is  a  tendency  to  crowd  M  and /together,  which  would 
be  so  pronounced  by  the  time  E  was  made  3"  long  as  to  be  altogether  inadmis- 
sible. Therefore  some  other  basis  for  establishing  the  length  of  the  E  scale 
must  be  found.  Let  us  multiply  the  first  column  of  the  determinant  by  0.2; 
9 


I30 


CONSTRUCTION    OF    GRAPHICAL    CHARTS 


then  when  we  subtract  the  —0.8  in  the  second  column  and  third  row  we  shall 
have  i  in  the  first  column  and  the  equations  for  M,  after  multiplying  the 
second  column  by  —  i,  will  be  x  =  0.8  and  y  =  log.  2  +  log.  M,  thus  making  it 
possible  to  lay  off  M  from  the  10  scale  and  have  its  length  3".  The  successive 
steps  are  shown  below. 


.2 

O 

-log.E 

.O 

•4 

-log./ 

= 

.2 

-.8 

log.  2+  log.  IT 

.2 


=    -.4 


I      — 


-log.£ 

logj 

•4 
-.8     log.  2+log.  M 

Then  the  construction  equations  are 


—  I 


-log.JS 

-log.  f 

log.  2+log.  M 
(x)  (y) 

-log.j 


x       ,  >i/ 

•4 

i      .8     log.  2+log.  M 


(E) 


0.2 


=    I 
_   log-/ 


(Jf) 


y  = 


o  .4 


x  =  0.8 

y  =  log.  2+log.Af 


Therefore  E  is  to  be  plotted  downward  on  the  Y-axis  from  the  20  scale, 
/  is  to  be  plotted  upward  on  a  scale  one  unit  to  the  right  of  F,  and,  at  0.8 
of  this  distance,  the  vertical  scale  for  M  is  located.  The  40  scale  is  to  be  used 
for/  and  the  10  scale  for  M. 

In  work  of  this  character  I  generally  find  it  convenient  to  ignore  the 
position  of  the  X-axis  altogether.  I  place  the  two  outside  seales  (here  E 
and  /)  so  that  their  centres  are  about  opposite  each  other.  The  distance 
apart  is,  of  course,  purely  arbitrary,  and  in  this  case  is  made  2  1/2";  the 
M  scale  is  then  placed  2"  from  E  (0.8  X  2  1/2).  From  a  table  of  logarithms 
lay  off  the  desired  graduations  on  E  and  /,  using  the  plan  followed  on  the 
slide  rule  of  dropping  the  smaller  graduations  as  we  move  up  to  the  higher 
values.  The  M  equation  shows  that  the  zero  position,  where  M  =  i,  is 
at  a  distance  log.  2  above  the  X-axis,  but  as  we  know  nothing  about  the  loca- 
tion of  X  we  must  fix  the  position  of  the  M  scale  in  some  other  way.  Cal- 
culate two  or  three  values  of  M  corresponding,  let  us  say,  to  E  =  25,000,000 
and/  =  200,000,  E  =  30,000,000  and/  =  50,000,  and  E  =  35,000,000  and/ 
=  10,000.  What  we  really  want  is  not  M  but  the  logarithm,  hence  the  cal- 
culations had  better  be  made  by  logarithms.  For  the  three  conditions 
specified  above  the  logarithms  are  2.90309,  1.61979  and  0.15490.  Connect 
the  selected  points  on  E  and  /  and  mark  the  intersections  of  the  lines  with 
the  M  scale.  Lay  the  10  scale  along  this  line  and  shift  it  back  and  forth 


THE    USE    OF    DETERMINANTS  131 

until  the  marks  on  M  agree  with  the  scale  at  2.903,  1.620  and  0.155,  and  with 
the  scale  in  this  position  graduate  the  desired  points  on  M .  One  point  should, 
of  course,  be  sufficient  to  locate  the  scale,  but  if  there  are  several  they  will 
serve  as  a  check  on  each  other  and  on  the  rest  of  the  work.  The  final  result 
is  shown  in  Fig.  66.  To  read  the  chart  join  the  selected  points  on  E  and  / 
and  find  the  intersection  with  the  middle  scale  M. 


10 


FIG.  66. — Chart  for  modulus  of  resilience. 


A  FOUR  VARIABLE  CHART  FOR  THE  ENERGY  OF  A  MOVING  MASS. 

Where  more  than  three  variables  occur  in  an  equation  it  may  be  charted 
in  a  number  of  different  ways.  A  few  typical  arrangements  will  be  shown 
in  the  following  examples.  Let  the  first  equation  be 

W 

E  =  -(VS-VS) 

in  which  E  represents  the  energy  due  to  a  change  in  velocity  of  a  mass  W 
from  Vi  to  F2. 

Write  the  equation 

2gE  -  WVi2  +  WV22  =  o 


132 


CONSTRUCTION    OF    GRAPHICAL    CHARTS 


and    let    x  =  E,  or  x  —  E  =  o,   and  y  =  Fi2,  or  y  —  Fi2  =  o.    Then  by 
substitution  we  have  2gx  —  Wy  +  JFF22  =  o  and  the  determinant  will  be 

i         o         —E 

o          i      — Fi2  =o 


2g     -W    WVJ 

In  this  case  as  there  are  four  variables  and  only  three  rows  we  are  obliged 
to  break  the  rule  with  which  we  started  and  put  two  variables  in  one  row. 
To  get  the  determinant  into  shape  for  plotting  subtract  the  second  column 
from  the  first  and  divide  by  the  first.  Then  multiply  the  second  column 
by  —i. 

-E 


-i 


2g+W     -W      TFF22 


(E) 


x  =  o 


y=-E 


(*) 

(y) 

i          o            -E 

I              O 

-E 

" 

i      -i                Fi2 
-W  \     TFF22 

I          I 
W 

vs 

JFF22 

2g+W       2g  +  W 

2g+W 

2g+W 

nations  then  are 

(x       i                           (x 

W 

2g+W 

PFF22 

=  o 


They  show  that  E  and  F  are  -plotted  on  two  vertical  axes,  one  positive 
and  the  other  negative,  and  that  W  and  F  are  represented  by  a  network 
of  lines  between  them.  To  establish  the  scale  lengths  let  us  suppose  that 
FI  and  F2  may  vary  between  i  and  10,  and  that  W  has  the  same  limits.  E 
will  then  vary  from  o  to  about  15.  To  keep  within  the  usual  3"  limit  of 
length  E  should  be  divided  by  5  and  Fi2  (100)  by  40,  or  the  equations  should 

E  Fi2 

read  y  =  —  and  y  —  —  •    These  results  will  be  attained  if,  in  the  original 
5  4° 

determinant,  we  multiply  the  first  column  by  5  and  the  second  by  40,  and 
then  take  the  same  steps  as  before. 


5              o          —  E                 5                    o         —  E 

o            40         —  Fi2=     —40                  40         —  Fi2 

= 

lOg 

-4oTF      TFF22        iog+4o 
-E   - 

I                     0 

5 

IV 

-4oPF      TFF22 

{A 

I                     O 

(y) 

-E 

5 

i           —i 

40 

- 

I              I 

40 

-4oJF             TFF22 

4oTF 

JFF22 

iog+4oW       iog+4oW 

iog+4oW 

iog+4oW 

=  o 


THE    USE    OF   DETERMINANTS 


133 


Then 


x  =  o 


-E 


CFO 


Zi! 
40 


= 


The  arrangement  is  shown  in  Fig.  67  where,  since  E  and  Fi  have  opposite 
signs,  we  have  the  familiar  Z  chart  with  its  sloping  X-axis.  The  unit 
length  on  X  is  made  3  1/3  "  as  before;  E  is  plotted  downward  on  the  Y-axis 
from  the  50  scale,  and  Fi  is  plotted  upward  from  the  40  scale.  In  the 


0T10 


15-^-15 


FIG.  67. — Chart  for  the  energy  of  moving  mass. 

third  set  of  equations  x  is  a  function  of  W  only.  The  x  equation  may  be 
solved  for  the  desired  values  of  W  and  the  results  plotted  on  the  X-axis 
from  the  30  scale,  since  the  unit  length  on  X  is  100  on  the  30  scale,  or  we 
may  project  the  graduations  from  E  using  as  a  pole  the  point  marked  8 
on  Fi.  This  pole  is  found  by  trial  as  explained  before,  or  may  be  calculated 
by  making  W  =  E  and  F2  =  o  in  the  original  equation.  The  projection  of 
points  9  and  10  from  this  pole  is  shown  in  the  figure.  As  for  the  equa- 
TFF22 


tion  y 


let  us  give  W  some  fixed  value,  say  10,  and  solve  for 

ioF22 


log  +  4oJF 

the  desired  values  of  F2.     The  equation  will  be  y  = 
1000 


With  F2 


10,  ;y 


=  1.39 


320  -f-  400 
Similarly,  with  F2  equal  to  9,  8,  7,  6,  5,  4,  3,  2  and 


134  CONSTRUCTION   OF   GRAPHICAL   CHARTS 

i,  y  will  be  1.126,  0.889,  0.681,  0.5,  0.378,  0.222,  0.125,  0.055,  an(^  0.014. 
Draw  a  vertical  line  through  the  point  corresponding  to  W  =  10  on  X,  and 
on  it  lay  off  the  calculated  values  of  y  just  found,  in  inches.  Similar  calcula- 
tions may  be  made  for  other  values  of  W  and  the  points  plotted.  Then  if 
all  the  points  marked  F2  =  10,  all  marked  F2  =  9,  etc.,  be  joined  it  will  be 
found  that  they  lie  along  straight  lines  passing  through  the  origin.  This  could 
have  been  predicted  if  we  had  noticed  that  x  and  y  have  a  constant  ratio  for 
any  particular  constant  value  of  F2.  It  was  therefore  unnecessary  to  find  the 
values  of  y  for  more  than  one  value  of  W.  Furthermore  even  these  calcula- 
tions might  have  been  dispensed  with  since  y  may  be  projected  direct  from 
the  Vi  scale  with  the  origin  as  a  pole,  for  if  E  in  the  original  equation  is  made 
zero,  FI  =  F2.  The  projection  of  the  lines  for  F2  (corresponding  to  9  and 
10)  is  indicated  on  the  figure. 

Fig.  67  shows  that  W  is  represented  by  a  series  of  vertical  lines,  and  that  F2 
is  represented  by  a  series  of  lines  converging  on  the  origin.  To  use  the  chart 
find  the  point  of  intersection  of  the  lines  representing  W  and  F2  and  join  it 
to  Fi  on  the  axis  to  the  right.  This  line  prolonged  to  E  will  give  the  desired 
value  of  that  quantity. 

A  FOUR  VARIABLE  CHART  FOR  LAME'S  THICK  CYLINDER  FORMULA. 
For  the  next  four  variable  chart  take  the  Lame  formula  for  thick  cylinders 


t  =  r 


+P  _  ,\ 

-#    V 


where  /  is  the  thickness  of  the  cylinder,  r  its  inside  radius,  /  the  fiber  stress 
and  p  the  internal  pressure.     Write  this 


if 

In  order  to  get  /  and  p  from  under  the  radical  we  must  square  both  sides 
of  the  equation,  but  this  will  mix  up  the  first  and  second  powers  of  /  and  r 
in  a  very  undesirable  fashion.  If,  however,  we  write  R  —  r  for  /  (R  being  the 
outside  radius  of  the  cylinder)  the  equation  becomes 


'  r2       f-p 

Now  put  each  side  of  the  equation  equal  to  the  arbitrary  quantity  A. 
Then  — ^  =  A  or  R*  =  Ar*  or  R*  —  Ar2  =  o,  and {.  __       =  A  or/  +  p  = 

A(f-  p)orf+p-  A(f-p)  =o. 


THE   USE    OF   DETERMINANTS 


135 


In  the  first  of  these  equations,  R2  —  Ar2  =  o,  let  x  =  R2  or  x  —  R2  =  o, 
and  y  =  A  or  y  —  A  =  Q\  then  x  —  yr2  =  o.  From  these  equations  write  the 
determinant. 

i         o        -R2 

0  i        —A      —  o 

1  -r2        o 

Now  multiply  the  second  column  by  —  i,  add  it  to  the  first  and  divide  by 
the  first. 

(x)        (y) 


Therefc 
(*) 

I 

o  -R2 

T  A 

I              0      -R2 

T                        T                       A 

I            O 

I        I 
r 

-R2 
A 

2 

=  o 
r2 

I 

Dre 

* 

(y 

r2  o 
=  o 

i+r2     r2       o 

(A)    \X  =  l 
[y*  A 

r 

[   i  + 

r*       ° 

<"i; 

i+r2 

=   0 

Let  the  limits  for  R  and  r  be  10  and  6  respectively,  and  for  /  and  p 
6000  and  3000.  Using  the  maximum  values  for  R2  and  r2,  A  will  be  approxi- 
mately 3.  To  bring  .ft2  (  =  100)  within  the  customary  length  let  us  write 


its  equation  y 


40 


and  since  the  maximum  A  is  3  its  y  equation  may  stand 


as  it  is.     The  only  thing  we  need  to  do  to  the  original  determinant  therefore 
is  to  multiply  the  first  column  by  40  and  then  take  the  same  steps  as  before. 

(*)     (y) 


„_             ._             D2 

.  _           <~i              D2 

Ar.              _             r>2 

-£2 

40        o       J\ 
o        i    —A 

— 

40        O          /v 

o  —  i    —A 

= 

-i       -i     -A 

- 

40 

i       i         A 
r2 

40     —  r2       o 

40        T             O 

^.Q-\~T         T                O 

I                 i        o         O 

40  \f 

=  o 


From  this  write  the  construction  equations 


(R) 


x  =  o 
-R2 


(A) 


y  = 


(r) 


X  =  r- « 

40+r2 
y  =  o 


Plot  the  squares  of  R  downward  on  the  Y-axis  from  the  40  scale. 
Draw  the  sloping  X-axis  and  make  it,  as  before,  31/3"  long.  Calling  this  the 
unit  length  for  X  we  draw  a  vertical  at  the  end  of  this  line  for  A ,  making  it 


3"  long.     Calculate  x  = 


40  + 


for  as  many  values  of  r  as  are  wanted  and 


plot  them  on  X,  or  project  them  from  R  using  a  pole  one  inch  from  the  lower 


136 


CONSTRUCTION   OF   GRAPHICAL   CHARTS 


R2 
end  of  A.     (Because  with  the  equation  —^  =  A,  A  becomes  i  when  R  =  r.) 

A  is  left  ungraduated  since  we  are  not  concerned  with  the.  numerical  values 
represented  by  A  and  only  use  the  line  as  a  stepping  stone  to  the  quantities 
/  and  p. 

Now  take  the  second  half  of  the  equation  with  which  we  started,/  -f-  p  — 
A(f  —  p)  =o  and  in  it  let  x  =  f  +  p  and  y  =  f  —  p.  Since  each  of  these 
equations  contains  both  /  and  p  we  must  eliminate  first  one  and  then  the 
other  variable.  Adding  the  equations  we  have  x  -\-  y  =  2/,  or  x  -f  y  — 
2/  =  o.  Subtracting  we  have  x  —  y  =  2p,  or  x  —  y  —  2p  —  o.  Then  sub- 
stituting x  and  y  in  the  first  equation  we  have  x  —  Ay  =  o.  We  now  have 
three  equations  each  containing  a  single  variable,  which  may  be  used  in 
writing  the  determinant. 


i  —A        o 

I  I       -2/ 

I     —I     —  2p 


=  O 


This  determinant  contains  a  ready-made  column  of  ones,  and  will  need 
no  other  change  than  to  multiply  the  second  and  third  columns  by  —  i. 

(y)   (*). 

i      A       o 

I     —I         2/    =0 
I          I        2p 


The  construction  equations  therefore  are 


(A) 


x  =  o 


y  =  A 


(P) 


X  =   2p 


y  =    -i  (y  =  i 

The  idea  here  is  to  construct  another  chart  with  an  yl-axis  which  is 
identical  with  this  axis  in  the  first.  The  y  equation  for  A  is  the  same  in  each 
case,  and  it  should  be  evident  that  the  two  charts  may  be  joined  along  this 
line,  A  of  course  must  be  looked  upon  as  the  Y-axis  for  the  second  chart. 
For  convenience  the  X-axis  is  sloped  up  as  shown  in  Fig.  68.  The  equations 
for/  and  p  show  that  they  lie  on  parallel  lines  to  X,f  being  i"  below  and  p  i" 
above.  We  will  suppose  that  p  is  needed  between  o  and  3000  and/  between 
2000  and  6000.  The  choice  of  the  scale  in  which  to  lay  out  these  quantities 
is  purely  a  matter  of  convenience  so  long  as  it  is  the  same  for  each  of  them; 
it  has  no  relation  to  the  y  scale,  and  none  whatever  to  the  x  scale  used  in  the 
other  half  of  the  chart.  The  2  in  front  of/  and  p  has  therefore  no  significance 
and  may  be  disregarded.  In  Fig.  68  I  have  made  i"  represent  2000  for 
both  /  and  p. 


THE   USE   OF   DETERMINANTS 


137 


To  read  the  chart  join  the  selected  points  on/  and  p  and  prolong  the  line 
to  its  intersection  with  A.  From  this  point  draw  a  line  through  r;  its  inter- 
section with  R  will  give  the  solution  of  the  equation. 

On  this  chart  I  have  thought  it  advisable  to  illustrate  a  point  which  is 
often  of  .considerable  practical  value.  The  reader  has  doubtless  observed 
that  while  satisfactory  readings  may  be  made  on  the  Z  type  of  chart  around 
the  middle  of  the  scale  the  results  are  apt  to  be  poor  as  we  approach  the  lower 


.5 
1 

1.5 
2- 

2.5- 


3- 


-Y  Axis  for 
First  Chart 


FIG.  68. — Chart  for  strength  of  thick  cylinders. 

limits.  If  both  large  and  small  readings  are  likely  to  be  wanted  it  will  pay 
to  doubly  graduate  some  of  the  scales,  that  is,  after  arranging  them  so  as  to 
take  in  the  maximum  values  we  will  enlarge  or  magnify  those  portions  lying 
at  the  lower  ends.  In  the  present  instance  it  will  be  seen  that  if  we  are 
lining  up  with  a  point  at  the  lower  end  of  A  and  a  point  on  r  near  or  below  2 
the  reading  on  R  will  be  decidedly  unsatisfactory.  Suppose  therefore  that 
we  expand  the  portion  of  the  scale  occupied  by  3  on  R  to  a  length  of  3", 


138  CONSTRUCTION  OF   GRAPHICAL   CHARTS 

and  graduate  for  every  tenth  of  an  inch  of  R,  the  result  being  shown  on  the 
left  hand  side  of  the  R  line.  These  graduations  may  be  projected  as  before 
to  the  r  scale,  and  from  the  same  pole.  The  readings  from  these  two  new 
scales  will  be  very  much  more  accurate  than  was  possible  with  the  first 
arrangement.  It  should  be  unnecessary  to  say  that  the  r,R  readings  must 
be  made  entirely  on  the  new  scales  or  entirely  on  the  old,  it  will  not  do  to  mix 
them.  The  possibility  of  making  a  mistake  here  is  the  only  objectionable 
feature  of  this  arrangement.  To  avoid  this  as  far  as  possible  the  scales  which 
are  to  be  paired  should  be  lettered  or  numbered  in  some  distinctive  manner. 
Here  I  have  used  heavy-faced  figures  for  one  pair  of  scales  and  light-faced 
figures  for  the  other.  Another  way  of  distinguishing  them  is  to  enclose  one 
pair  of  mating  figures  in  parentheses. 


FOUR  OR  MORE  VARIABLE  LOGARITHMIC  CHARTS. 

An  equation  containing  four  variables  and  of  the  simple  multiplication 
type  will  generally  be  best  represented  by  a  chart  with  logarithmic  scales. 
If  for  instance  the  equation  reads  ab  =  cd  we  may  let  ab  and  cd  each  equal 

a   c     A     d   b  a       k        A     c  ,  rf 


FIG.  69. 


FIG.  70. 


A,  and  construct  two  charts  with  an  "A  "  dummy  axis  in  each,  and  let  these 
A  -axes  coincide  as  was  done  in  the  last  problem.  A  may  lie  between  both 
a  and  b,  and  c  and  d  as  sketched  in  Fig.  69,  or  it  may  be  outside  of  one  or  of 
both  pairs  as  in  Fig.  70.  In  any  case  the  index  lines  for  the  two  pairs  must 
intersect  on  A.  Where  the  arrangement  shown  in  Fig.  69  is  used  some  authori- 
ties advise  that  the  a  and  c  lines  be  made  to  coincide  and  also  b  and  d,  each 
scale  occupying  one  side  of  a  line.  My  own  feeling  is  that  this  is  distinctly 
bad,  for  even  with  the  closest  attention  errors  will  be  made  through  con- 
necting up  the  wrong  scales. 

An  arrangement  for  five  variables  is  shown  in  Fig.  71.  Here  the  result 
of  the  operations  on  a  and  b  is  found  on  A ,  while  for  c  and  d  it  is  found  on  B. 
Then  by  joining  the  points  found  on  A  and  B  we  get  the  final  result  on  e. 
With  six  variables  we  might  have  three  ungraduated  dummy  axes,  A,  B 
and  C,  Fig.  72,  A  carrying  the  result  of  the  operation  on  a  and  b,  B  that  on 


THE   USE   OF   DETERMINANTS 


139 


c  and  d,  and  C  that  on  e  and  /,  and  finally  the  points  found  on  A ,  B  and  C 
must  lie  on  one  straight  line.  An  example  of  the  type  shown  in  Fig.  69  will  be 
worked  out,  the  general  method  used  for  this  being  the  same  for  all  of  them. 

aAb        c        cBd  aAbcBdeC* 


FIG.  71. 


FIG.  72. 


A  FOUR  VARIABLE  LOGARITHMIC  CHART  FOR  HELICAL  COMPRESSION 

SPRINGS. 

The  formula  chosen  will  be  that  for  the  strength  of  helical  compression 
springs 

p  = 


where  P  is  the  load,  d  the  diameter  of  the  wire,  /  the  fiber  stress  and  r  the 
radius  of  the  coil.  Write  this 

Pr  =  A  =  O.IQ63J3/ 

Then  log.  P  +  log.  r  —  log.  A  =  o,  and  with  x  =  log.  P,  and  y  =  log.  r  we  have 
by  substitution  x  +  y  —  log.  A  =  o.  From  these  equations  the  determinant 
becomes 

i       o    —log.  P 


i     —log.  r 


=  o 


i       i    —log.  A 

Add  the  second  column  to  the  first,  divide  by  the  first  and  multiply  the 
third  by  —  i 

(*)     60 


I 
i 

o     -log.P 
i      —log.  r 

- 

i 
i 

o     -log.P 

i     -log.  r 
i     —log.  A 

— 

i 
i 

o    log.P 

i     log.  r 
i     log.  A 

=  0 

Whence 
d 

2 

')' 

i         log.  A 
x  =  o 

i 
W 

2               2 
X=I 

i 
(A) 

2             2 

I 

X=~ 
2 

_   ^g.  ; 

4 

y    1Qg-  * 

y     log.  r 

y                2 

Assuming  the  limits  forP  to  be  from  i  to  200  and  r  from  1/2  to  2,  and  that 
the  outside  scales  are  to  be  about  3"  long,  as  before,  we  adjust  the  logarithms 


I4O 


CONSTRUCTION   OF   GRAPHICAL   CHARTS 


of  these  variations  to  this  length.  The  logarithm  of  200  is  2.301  and  of  i 
is  o,  a  total  logarithmic  variation  of  2.301.  This  will  be  near  enough  to  3 
to  stand  without  any  modification,  and  the  equation  y  =  log.  P  will  be  left  as 
it  is.  The  logarithm  o£  2  is  0.301,  while  that  of  0.5  is  1.699,  a  difference  of 
0.602.  If  this  were  divided  by  0.2  we  should  have  about  3,  so  let  us  say  that 

- 


y— 


This  change  will  be  made  if  the  second  column  of  the  first  determi- 


nant is  multiplied  by  0.2.     Then  we  take  the  same  steps  as  before. 

(*)    (y) 


i      o  -log.P 

i  o  —log.P 

i     o      —log.P 

I       0        log.  P 

o     .2   —log.  r 
i     .2   —log.  A 

— 

.2  .2  —log.  r 

1.2  .2  —  log.  ^4 

=» 

-log.r 

= 

log.r 

i     i 

.2 

i     —log.  A 

i     i 

.2 

i     log.  A 

1     6          1.2 

1     6       1.2 

Therefore 

x=o 

X=I 

I 

(P)    < 

(r) 

y  =  \Qg.P 

y 

(A) 

_  log-  ' 

log-  A 

.2 

1.2 

=  0 


From  these  equations  it  is  seen  that  P  is  to  be  plotted  from  the  10  side  of 
the  scale  and  r  from  the  20  side.  Since  the  dummy  A  scale  is  not  to  be  gradu- 
ated its  odd  1.2  unit  need  not  bother  us;  it  is  merely  necessary  to  see  that  the 
A  equation  in  the  second  half  of  the  chart  agrees  with  this  A  equation. 

The  second  half  of  the  original  equation,  A  =  0.1963^7,  may  be  written 
log.  A  —  log.  0.1963  —  3  log.  d  —  log.  /  =  o.  Then  letting  x  =  log.  d  and  y  = 
log.  /  and  substituting  we  have  log.  A  —  log.  0.1963  —  32  —  y  =  o.  Write 
the  determinant  as  usual,  add  the  second  column  to  the  first,  divide  by  the 
first  and  multiply  the  third  by  —  i. 


I 
0 

-3 

o               —log.  d 
i               -log./ 
—  i     log.  A—  log.  0.1963 

i     6                    -^log.  </ 

i     i                     -log./ 
i          /log.  A—  log.  o. 

[Qf 

3\ 

I 
i 
4 

o                —log.  d 
i                -log-/ 
—  i      log.  A—  log.  0.1963 

(*)             (y) 

I       O                          log.  J 
I        I                            log./ 

i     log.  A—  log.  0.1963 

ii                                   i 

4          \                 4                 / 

4                   4 

=  o 


Then 


(d) 


=  \o.  d 


x  —  i 


(A) 


_log.  A  —  log.  0.1963 
4 


THE   USE    OF   DETERMINANTS  141 

The  limiting  values  for  d  will  betaken  as  0.05  and  0.25,  and  for/  20,000  and 
100,000.  The  first  thing  to  attend  to  is  getting  the  A  equation  to  agree  with 

that  in  the  first  chart  where  y=  -     - —    The  appearance  of  the  log.  0.1963 

in  the  y  equation  for  A  need  not  disturb  us  since  its  only  effect  on  the  A  scale  is 
to  locate  its  zero  position  from  X,  and  since  the  scales  will  be  located  without 
using  X,  as  was  done  in  Fig.  66,  we  may  disregard  log.  o.  1963  and  consider  only 

log.  A  log.  A  . 

y  =  — This  can  be  made  into  y  =  —      —  by  dividing  by  0.3.     If  this  were 

done  by  dividing  the  third. column  of  the  last  determinant  by  0.3  we  should 
have,  instead  of  the  given  equations  for  d  and  f,y=~  -  and  y=  -  — » 

the  x  equations  being,  of  course,  unchanged.  Let  us  see  if  this  can  be  made 
to  do.  Consider  d  first;  the  logarithm  of  0.25  is  1.398,  and  of  0.05  is  2.699, the 
difference  being  0.699.  M  this  is  divided  by  0.3  we  shall  have  a  length  of  2.33" 
which  is  near  enough  to  the  limiting  length  of  3"  to  answer.  With/  varying 
between  100,000  and  20,000,  the  logarithms  of  which  are  5  and  4.301,  we  again 
have  a  difference  of  0.699  and  a  scale  length  of  2.33''.  This  being  considered 
satisfactory  and  no  changes  having  been  made  in  the  x  equations  we  have 
no  further  need  to  bother  with  the  determinant. 

In  this  case,  as  was  stated  above,  no  attention  whatever  will  be  paid  to 
the  location  of  the  X-axis;  it  is  frequently  not  on  the  paper  at  all.  The  x 
equations  give  the  right  spacing  for  the  vertical  scales  as  a  proportion; 
we  are  not  even  obliged  to  use  the  same  scale  unit  for  x  in  the  first  and  second 
charts.  For  instance,  A  divides  the  distance  between  P  and  r  in  the  pro- 
portion of  1/6  to  5/6,  and  the  distance  between  d  and/  in  the  proportion  1/4 
to  3/4.  In  the  chart  shown  in  Fig.  73  the  distance  between  P  and  A  has  been 
made  3/8",  and  between  A  and  r,  i  7/8".  Then  to  get  sufficient  separation 
between  P  and  d  the  latter  is  put  3/4"  from  A,  while  /  is  placed  2  1/4"  on 
the  other  side  of  A . 

As  regards  the  positions  of  the  scales  on  the  vertical  axes  three  of  them 
may  be  anywhere  we  please,  and  for  the  sake  of  appearance  should  be  as 
nearly  opposite  each  other  as  possible.  In  the  interest  of  accuracy  those 
farthest  from  the  A  -axis  should  be  plotted  first.  The  positive  signs  of  the 
equations  show  that  all  of  the  scales  are  to  be  plotted  upward.  The  30 
scale  is  to  be  used  for  d  and/,  the  20  scale  for  r  and  the  10  scale  for  P.  From 
a  table  of  logarithms  lay  off  the  desired  graduations  for  d,  r,  and/,  arranging 
them  so  that  the  middle  of  each  scale  shall  lie  on  a  horizontal  line  across  the 
middle  of  the  sheet.  Then  draw  index  lines  to  indicate  the  solution  of  the 
equation  for,  say,  three  sets  of  conditions.  For  instance  let  d  .=  0.25,  /  = 


142 


CONSTRUCTION    OF    GRAPHICAL   CHARTS 


100,000  and  r  =  2;  d  =  o.i,/  =  50,000  and  r  —  i;  d  =  0.05,  /  =  20,000  and 
r  =  0.5.  Join  d  and/  and  mark  the  intersection  with  A,  and  through  this 
point  and  r  draw  a  line  to  its  intersection  with  P  and  mark  it.  Calculate 
the  logarithms  of  P  for  the  three  given  sets  of  conditions.  They  will  be 
2.186,  0.992  and  1.991.  Then  lay  the  10  scale  along  the  P-axis  and  adjust  it 
till  the  marks  on  the  paper  coincide  with  these  numbers  on  the  scale.  It 
is  then  in  the  correct  position  and  we  may  proceed  to  graduate  the  entire 
scale.  The  final  result  is  shown  in  Fig.  73. 

To  read  the  chart,  it  is  only  necessary  to  see  that  the  lines  joining  the 
selected  points  on  the  outer  and  inner  scales  intersect  A  at  the  same  point. 


.25-r;25 


200-,-200 


.05 


FIG.  73. — Chart  for  strength  of  helical  springs. 

This  chart  may  be  profitably  compared  with  the  chart  for  the  same  equation 
shown  in  Fig.  18. 

Exactly  the  same  general  method  is  to  be  used  if  the  chart  form  is  that 
shown  in  Fig.  70  except  that  the  determinant  must  be  so  arranged  as  to  put 
A  on  one  of  the  outside  axes  instead  of  between  them. 

A  FOUR  VARIABLE  Z  TYPE  CHART,  FOR  HELICAL  COMPRESSION  SPRINGS. 

As  a  matter  of  interest  let  us  plot  the  same  equation  on  the  Z  type  of 
chart.  Starting  as  before  with  the  equation 

pr  =  A  =  o.i963</3/ 


THE    USE    OF    DETERMINANTS 


143 


write  the  first  part  Pr  —  A  =  o.  Let  x  =  P  and  y  =  A ,  then  rx  —  y  =  o. 
Form  the  determinant,  subtract  the  second  column  from  the  first,  divide 
by  the  first  and  multiply  the  second  by  —  i. 


I          0     -P 

o      i   —A 
r  —i        o 

Therefore 

»• 

I                0      -P 

-i           i    -A 

r-f-i     —i         o 

- 

I         0         -P 

i  -i         A 
—  i 

I/-* 

= 

I          0        -P 

i       i         A 

i 

1                    ° 

r+i 

I          .                O 

r-fi 

X  =  O 


X  =   I 


X  = 


«  W) 

ly  «  — P  ly  =  4  ly  =  o 

These  equations  show  that  P  and  ^4  are  to  be  plotted  on  vertical  axes, 
while  r  lies  on  the  sloping  X-axis  between  them.  Let  P  and  r  have  the  maxi- 
mum values  of  150  and  2;  then  Pr,  or  A,  will  have  a  maximum  value  of  300. 

To  bring  P  within  the  usual  3"  length  write  its  equation  y  = The 

A 

same  result  may  be  accomplished  for  A  by  writing  its  equation  y  =  -  — 

In  the  first  determinant  multiply  the  first  column  by  50  and  the  second  by 
loo.  (The  final  operation  above  will  be  saved  if  we  make  this  — 100  and 
add  the  second  column  to  the  first  instead  of  subtracting).  Then  divide 
by  the  first  column. 


o   -P 


5° 

o    -P 

o 

—  100    —A 

= 

S°r 

IOO            0 

=     —IOO 


—  IOO    —A    = 


50^+100       100 


= 

I                0 

I            I 

IOO 

T 

(y) 
-p 

1 

IOO 
0 

5or-fioo 

=  o 


Then 


(P) 


X  =  O 

X  =   I 

-p    (A} 

A 

to 

'V  ^  

v  ==  —  " 

50 

IOO 

IOO 


10 


50^+100       5H-io 


Lay  off  P  downward  on  the  Y  axis  using  the  50  scale.  At  one  unit 
(here  3  1/3"  or  100  on  the  30  scale)  to  the  right  along  X  draw  a  vertical 
line  3"  long  to  represent  A.  This  is  not  to  be  graduated.  On  the  sloping 

10 


X-axis   r   is   to   be  graduated  from   the  equation  x 


+  10 


It  will 


be  noted  that  although  the  origin  from  which  x  is  measured  is  the  point 


144 


CONSTRUCTION  OF  GRAPHICAL  CHARTS 


where  P  intersects  X,  this  point  does  not  correspond  with  r  =  o  but  r  = 
oc .  When  r  =  o,  x  =  i .  This  point  is  the  foot  of  the  A  scale.  Hence 
r  cannot  be  graduated  by  projection  from  P,  but  could  be  projected  from  A 
if  we  were  to  lay  off  a  temporary  scale  on  that  line  for  this  purpose. 

Now  take  the  second  half  of  the  equation  and  write  it  A  —  0.1963^7  =  o. 
Then  letting  x  =  A  and  y  =  d3,  we  have  by  substitution  x  —  0.1963^7  =  o- 
Form  the  determinant,  subtract  the  first  column  from  the  second  and  divide 
by  the  second. 


i 

0 

-A 

0 

i 

-d* 

= 

I 

-.19637 

o 

Then 

—  I 


=   o 


-A 

-d* 


-(i  +  .i9637) 


(*) 

—  i 

o 

—  i 


(y) 

A 


I+.  19637 


U) 


x  =  —i 


(d) 


X  =  O 


y  =  -dz 


i     — i 


—  i 
••19637 


=  0 


We  must  first  bring  the  y  equation  for  A  into  agreement  with  that  used 
for  the  first  half  of  the  chart.  This  may  be  done  by  multiplying  the  first 
column  of  the  original  determinant  by  100.  Next,  with  a  maximum  value 
of  0.25  for  d  we  have  d*  =  0.015625,  and  this  divided  by  0.005  w^l  giye  us 
approximately  3"  for  the  maximum  length  of  y.  To  make  this  change  multi- 
ply the  second  column  by  0.005.  Then  proceed  as  before. 


100 

o 
100 


o  -A 

.005         -d* 

.0009817        o 


IOO 

o 

IOO 


—  IOO 


.005 

—  (ioo+.oooq8i7) 


—  i 


—  IOO 


-A 

-d* 
o 

Ml 

A 

IOO 


.005 


100+ .  0009817 


=  o 


Then 


X  = 


—  IOO 


100+ .  0009817 
y  =  o 


These  equations  show  that  d  is  plotted  downward  on  the  Y-axis,  and  that 
A  is  on  a  vertical  line  one  unit  to  the  left  of  d  measured  along  X.  Since  the 
A  equations  for  the  two  halves  of  the  chart  are  the  same,  the  lines  representing 


THE    USE    OF    DETERMINANTS 


145 


them  may  be  brought  into  coincidence.  For  the  sake  of  symmetry  make  the 
unit  length  on  X  and  the  slope  of  the  X-axis  the  same  on  both  sides  of  A. 
d  is  to  be  laid  downward  from  X,  as  a  scale  of  cubes,  from  the  50  side  of 
the  enginner's  scale,  while  /  is  to  be  graduated  along  X  according  to  the 
—  TOO 


formula  x 


100 +  0.0009817 


It  may,  if  desired,  be  projected  from  a  tem- 


150-L150 


.05 


.25-L25 


FIG.  74. — Chart  for  strength  of  helical  springs. 


porary  scale  on  A,  as  was  r,  since  the  zeros  for/ and  A  coincide.  The  result- 
ing chart  is  shown  in  Fig.  74.  It  is  read  by  joining  d  and/  and  extending  the 
line  to  A.  From  this  point  on  A  a  line  is  drawn  through  r;  its  intersection 
with  P  is  the  solution  sought. 

The  method  just  explained  is  capable  of 
extension  to  five,  six  or  any  number  of  vari- 
ables. If,  for  instance,  we  have  the  product 
abed  =  e  we  may  say  that  ab  =  A,  and  con- 
struct a  chart  with  A  on  an  outside  axis. 
Then  with  Ac  =  B  construct  another  with  A 
and  B  as  outside  axes,  and  joined  to  the  first 

along  A.     Finally  with  Bd  =  e  we  construct  a  third,  joined  to  the  second 
along  B,  the  general  scheme  being  shown  in  Fig.  75. 
IQ 


FIG.  75. 


146 


CONSTRUCTION   OF    GRAPHICAL    CHARTS 


A  PARALLEL  ALINEMENT  CHART  FOR  THE  BAZIN  FORMULA. 

I  wish,  next,  to  show  how  the  parallel  alinement  chart  described  in 
Chapter  V  may  be  handled  by  the  determinant  method.  Imagine  a  chart 
such  as  is  sketched  in  Fig.  76,  where  the  variables  a,b 
and  c, d  line  up  at  a  point  on  the  axis  A .  If  A  is  removed 
to  infinity  it  should  be  clear  that  the  two  index  lines  will 
be  parallel.  Knowing  a,  b  and  c,  for  instance,  we  join 
a  and  b}  and  draw  a  parallel  to  this  line  through  c.  Its 
intersection  with  d  will  be  at  the  required  value  of 
that  quantity.  As  an  illustration  of  this  type  let 
us  use  the  Bazin  formula  for  the  flow  of  water  in  open 
FlG-  76.  channels 


i  + 


N 


VR 


V 


/—    •<•«.     -,-r 

^S  .+-£= 


and  write  it 


separating  the  variables  in  such  a  manner  that  two  of  them,  V  and  S,  are 
placed  on  one  side  of  the  equality  sign  and  two,  R  and  N,  on  the  other. 
Then  write  the  first  half  of  the  equation 


Letting  x  =  V  and  y  =  A  we  have  after  substituting  x  —  y  \/S  =  o. 
Write  the  determinant  as  usual  and  divide  by  the  first  column. 


0  -V 

1  -A 

-Vs    o 


(*)     GO 

o        -V 

oc        —  oc 

-Vs     o 


Then 


=  -v 


y=° 


[y=- 


Take  as  the  limits  for  R,  o  to  10;  for  S,  o.oooi  to  o.i;  and  for  N,  o  to  3.2. 
Then  the  maximum  for  V  will  be  about  160.     If  160  is  divided  by  50  we 

—  V 
shall  have  about  the  right  length  for  the  scale,  therefore,  let  us  say  y  — 

With  the  maximum  value  of  o.i  for  S,  its  square  root  will  be  0.316.     This 


THE    USE    OF   DETERMINANTS  147 

divided  by  0.2  gives  1.58  which  will  be  supposed  to  be  satisfactory  for  the 

-Vs 

horizontal  scale  length.     Therefore  the  equation  will  be  x  =  —      —  • 

0.2 

These  results  will  be  attained  if  we  divide  the  second  column  of  the  last 
determinant  by  0.2  and  the  third  by  50. 

Now  write  the  second  half  of  the  equation 


Then  if  x  =A  and  y  =  AN=xN,  we  have  by  substitution  xVR  +  J  — 
i$$R  =  o. 

Write  the  determinant  from  these  equations  and  divide  by  the  second 
column. 

(*)  (y) 


i        o          —A 

-N     i  o 

VR    i     -158/2 


00  I  —   OC 

-N     i  o 

VR    i     -158* 


Then 

=  oc  (x  =  -N  (x  =  VR 

(N)  \  (R)\ 

y   =    —    oc  [y   =   Q  (y  =    ~I$oR 

If  R  =  10,  x  =  3.16  and  y  =  —1580;  while  for  N  =  3.2,  x  =  —3.2. 
The  scale  units  used  for  x  and  y  in  the  second  half  of  the  chart  need  not  be 
the  same  as  those  used  in  the  first  half,  but  the  ratio  between  the  x  and  y 
units  must  be  the  same  in  each  case,  otherwise  the  index  lines  will  not  be 


—  v 

parallel.     In  the  first  case  we  have  y  =  -    -  and  x  =  ~      —>  the  ratio  be- 

50  0.2 

tween  the  denominators  being  250.  In  the  second  case  we  have  —1580 
for  the  maximum  value  of  y  on  the  R  scale,  and  dividing  this  by  500  we  shall 
have  about  our  usual  3"  length.  To  have  the  ratio  of  the  denominators 
equal  to  250,  that  used  for  the  x  measurements  must  evidently  be  2.  There- 
fore the  equations  will  read  y  = >  x  —  -  -  and  x  = •  The 

500  2  2 

maximum  values  of  x  for  AT  and  R  will  be  —1.6  and  1.58,  a  total  width  for 
this  chart  of  3.18.  To  get  these  equations  divide  the  first  column  of  the  last 
determinant  by  2  and  the  third  by  500. 

The  X-  and  Y-axes  may  be  perpendicular  or  not  as  we  please,  and  although 
these  axes  for  the  two  charts  need  not  coincide  they  must  be  parallel.  In 
the  present  case  the  axes  will  be  made  rectangular  but  not  coincident,  as  it 
tends  to  some  confusion  if  two  scales  are  graduated  on  the  same  line. 

Draw  horizontal  and  vertical  lines  to  represent  the  X-  and  Y-axes  of  the 
first  half  of  the  chart.  On  the  X-axis,  and  to  the  left  of  the  origin,  lay  off 


148 


CONSTRUCTION   OF    GRAPHICAL   CHARTS 


such  values  of  x  = 


-V~s 


0.2 


as  are  needed  between  S  =  o.oooi  and 


o.i, 


using  a  table  of  square  roots  and  the  twenty  side  of  the  engineers'   scale. 

—  V 
On  the  Y-axis,  and  downward  from  the  origin,  lay  off  y  =  from  the  50 

scale.     Then  draw  another  pair  of  axes  parallel  to  the  first.     On  the  new 

—  N 
X-axis  and  to  the  left  of  the  origin,  lay  off  x  =  >  using  the  20  scale.     Then 


1 

1    1    1 

55 

CO                   1         <N                            -<                             0 

CO                            o3                            •""*      r4 

®.                                    3.                 §.    § 

I        ,     I        i        |          .              1.,  ..  i  .  ,   ,1,1 

N^ 

1-J 

S         *           S           | 

10- 

20- 
30- 

-10 

-20 
-  30 

N  \ 

\ 

\    fj^ 

*\ 

V 

40- 
60- 
60- 
70- 
80- 
90- 

v  —  ^ 

100- 
110- 
120- 
130- 
140- 
150- 
160- 

-  40 
-  50 
-60 
-70 
-  80 
-90 
-100 
-110 
-120 
-130 
-140 
-150 
-160 

4\ 

V 

i 

R~\* 

\ 

\  ^ 

6  \ 

A*1 

\ 

\* 

A 

A  c, 

l 

A 

FIG.  77.  —  Chart  for  Bazin  formula.] 

downward  on  the  Y-axis  lay  off  y  =  -         -  or  —0.316^.     As  we  have  no 

ready-made  scale  to  fit  this  equation  we  must  take  the  maximum  length 
which,  for  R  =  10  is  3.16",  and  divide  it  into  ten  and  as  many  smaller  equal 
parts  as  are  wanted,  by  any  of  the  methods  which  are  familiar  to  draftsmen. 
From  these  points  draw  horizontals  to  the  right  and  lay  off  on  them  the  values 


of  x 


corresponding  to  these  points,  and  join  by  a  smooth  curve.     See 


Fig.  77.     To  read  the  chart  join  R,  on  the  curve,  to  ^V,  and  parallel  to  this 
line  draw  one  through  S.     The  second  line  will  cut  V  at  the  desired  value. 


THE  USE  OF  DETERMINANTS  149 

SPECIAL  METHODS  FOR  REFRACTORY  EQUATIONS. 

Sometimes  a  refractory  equation  will  be  encountered  which  resists  all 
attempts  to  separate  the  variables  in  the  manner  which  has  been  outlined. 
It  is  not  always  the  equation  of  complex  appearance  which  gives  the  most 
trouble.  Often  such  equations  will  fall  into  their  component  parts  almost 
at  a  touch,  while  in  others  of  far  simpler  appearance  the  variables  will 
cling  together  in  a  most  exasperating  manner. 

Two  methods  of  handling  such  cases  are  open  to  us.  If  the  operation 
involved  is  some  simple  arithmetical  process,  such  as  addition,  subtraction, 
etc.,  which  is  easily  performed,  we  may  leave  this  to  be  done  by  the  user  of 
the  chart,  and  let  him  enter  it  with  the  result  of  the  calculation  instead  of 
the  separate  variables. 

Or  we  may  use  the  so-called  method  of  condensed  points. 

Neither  of  these  methods  is  desirable  as  a  general  rule,  but  they  will  some- 
times save  the  day  when  every  other  expedient  has  been  tried. 

CHART  FOR  SPHERICAL  DISTANCE. 

The  formula  chosen  to  illustrate  the  first  type  of  chart  is  for  what  is 
called  the  spherical  distance  between  two  points  on  the  earth's  surface  when 
their  latitudes  and  longitudes  are  known. 

2  cos.  ^4  =  (i+cos.  L)  cos.  (a— b)  —  (i  —  cos.  L)  cos.(a+£) 

Here  A  is  the  spherical  distance;  L,  the  difference  in  longitude;  and  a  and 
b  the  latitudes.  In  this  case  the  longitudes  of  the  two  points  are  not  shown 
separately  in  the  chart;  their  difference,  L,  is  first  calculated  and  the  result 
used  in  entering  the  chart.  Similarly  the  chart  will  show  the  quantities 
a— b  and  a-\-b  on  the  scales  instead  of  a  and  b  separately,  and  it  will  be  neces- 
sary to  perform  the  subtraction  and  addition  beforehand. 

Rewrite  the  equation 

2  cos.  A—  cos.  (a  —  b)—  cos.  Lcos.(a  —  £)+cos.(a-f-£)  —  cos.  L  cos.(a+&)  =  o 
2  cos.  A—  [cos. (a  —  b)—  cos.(a+&)]  —  cos.  L  [cos.(#  —  £)+cos.(a-f-£)]  =  o 

Let  x  =cos.(a— b)  —  cos.  (a+b)  and  y  =cos.(a  —  b)  +  cos.(0+£);  then 
adding  and  subtracting  these  equations  we  have  x-\-y=2  cos.  (a—b)  and 
x—y  —  —  2  cos.  (a+5).  Substituting  in  the  original  equation  we  have 
2  cos.  A— x  —  y  cos.  L  =  o.  The  last  three  equations  are  now  used  to  form 
the  determinant.  Divide  the  third  column  by  —2  and  then  divide  the 
determinant  through  by  the  first  column. 


CONSTRUCTION   OF    GRAPHICAL    CHARTS 


i  i         —  2  cos.  (a— b) 

i          —i  2  cos.(a+#) 

I          —  COS.Z,  2  COS.  A 


I 

—  I 
—  cos.L 


cos.(a-b) 
cos.(a 
—  cos.^4 


i 

—i 
cos.  L 


(y) 

cos.  (a  —  b) 
—  cos.  (a+b) 
cos.  ^4 


=  o 


The  construction  equations  taken  from  this  are 


f          ,.    (  X=I 

(a—b)  < 

J  \y  =  c( 


.  (a-b) 


x=—i 

y=  -cos.  (o+i) 


These  equations  indicate  that  the  quantity  cos.  (a  —  b)  is  to  be  plotted 
on  a  vertical  line  at  unit  distance  to  the  right  of  the  origin;  that  cos.  (a  +  b) 
is  to  be  plotted  in  the  opposite  direction  (on  account  of  its  negative  sign)  on  a 
vertical  axis  at  unit  distance  to  the  left  of  the  origin;  and  that  the  third 
point  has  as  coordinates  cos.  L  and  cos.  A. 

There  is  no  question  here  of  modifying  individual  scale  lengths  —  what 
is  done  to  one  is  done  to  all  for  either  x  or  y.  The  x  and  y  lengths  may  be 
different  if  we  wish  to  make  them  so,  but  as  there  will  be  no  advantage  in 
doing  this  we  will  make  them  the  same  and  get  a  square  chart.  In  this 
case  we  will  make  the  length  representing  unity  i  2/3"  which  may  be  divided 
decimally  by  the  use  of  the  60  scale. 

At  this  length  to  the  right  of  the  origin  draw  a  vertical  and  lay  off  on  it  a 
scale  of  cosines.  For  (a  —  b)  =  o,  y  =  i;  for  (a  —  b)  =  90°,  y  =  o;  and 
for  (a  —  b)  =  1  80°,  y  =  —i.  Then  going  one  unit  to  the  left  of  the  origin 
draw  another  vertical  and  lay  off  on  it  another  cosine  scale  for  (a  +  b)  which, 
on  account  of  the  negative  sign  will  have  o°  at  the  bottom  and  180°  at  the 
top.  These  are  the  two  outside  scales. 

The  intermediate  points  with  which  they  are  to  be  alined  are  found  by  a 
system  of  x  and  y  coordinates.  The  abscissas  are  represented  by  cos.  L  which 
the  equation  shows  are  to  be  graduated  along  the  X-axis  from  —  i  (for  cos. 
1  80°)  to  H-  1  (for  cos.  o°).  Through  these  points  verticals  are  drawn  for  the 
ordinates.  The  ordinates  are  represented  by  y  =  cos.  A  ,  which  are  graduated 
on  Y  and  carried  across  the  chart  by  horizontals.  Since  the  y  for  cos.  A  is 
identical  with  the  y  for  cos.  (a—b),  the  graduations  already  made  for 
the  latter  on  the  right  hand  scale  may  be  utilized  for  drawing  the  hori- 
zontals instead  of  repeating  them  on  Y.  The  finished  chart  is  shown  in 
Fig.  78. 

To  use  it  find  L,  (a  +  b)  and  (a  —  b).  Join  (a  +  b)  on  the  left  hand 
scale  with  (a  —  b)  on  the  right  and  find  the  point  of  intersection  of  this  line 


THE    USE    OF    DETERMINANTS 


with  the  vertical  through  L.  A  horizontal  through  this  point  will  intersect 
the  A  scale  (which  is  identical  with  (a  —  b)  at  the  right)  at  the  desired  value 
of  that  quantity.  For  instance  with  (a  —  b).=  60°,  (a  +  b)  =  30°  and 
L  =  40°,  A  =  70°. 

.180     ICO  140    130     120      110       100  \    SO        80          70        60       50     40    30  20  9 


on 

160 

OA 

150 

40 

140 

rA 

130 

190 

60 

/ 

S' 

70* 

/ 

80 

-  -a 

-b 

X  i 

i 

LxlS 

f-Y  ^ 

3     ' 

/ 

and 

b 
A 

00 

yo 

/ 

' 

X 

110 

70 

x 

x 

190 

CO 

x 

130 

r  50 

£ 

x 

110 

X 

150 

20 

ICO 

80        70       00       50     40    30  20  0 
FIG.  78. — Chart  for  spherical  distance. 


180160150140  130      120      HO       100  T    90 


CHART  FOR  BRAKE  BANDS. 

As  an  illustration  of  the  method  of  condensed  points  take  the  formula 
for  the  tension  in  the  tight  side  of  a  brake  band. 

pTnO. 0076/c 

T=r 

IO0.0076/'o j 

where  T  is  the  tension;  P,  the  resultant  tangential  pull  on  the  brake  drum; 
/,  the  coefficient  of  friction;  and  a  the  arc  of  contact  of  the  band. 

The  difficulty  here  is  to  separate  the  quantities  /  and  a  in  such  a  manner 
that  we  can  use  the  equations  in  writing  the  determinant.     Let  us  treat  the 

IQ0. 0076/0 

expression  — 0  0076/0_   '   which  is  a  function  of  both  /  and  a,  as  a  single 

variable,  and  for  simplicity  call  it  A.     Then  T  =  PA,  or  T  —  PA  =  o. 
Let  x  =  T  and  y  =  A ,  then  x  —  Py  =  o. 


152 


CONSTRUCTION    OF    GRAPHICAL    CHARTS 


Write  the  determinant,  subtract  the  second  column  from  the  first,  divide 
by  the  first  column  and  multiply  the  second  by  —  i. 


i         o     -T 

0  i     —A 

1  -P         o 


i  o     -T 

-i  i     -A 

i+P     -P        o 


. 

i          o     —  T 

i      -i        A 
-P 

= 

(*)     00 

i       o     —  T 

i       i         A 
P 

T                                 O 

1     i+P     ° 

=  o 


The  construction  equations  from  this  arc 


(T) 


y=  -T 


(A) 


(P) 


i+P 


y  =  o 


Let  the  limits  for  P  be  o  and  100;  for  T,  o  to  260;  for/,  0.15  to  0.3;  and  for 

T 

a,  1 80°  to  270°.     Then,  for  the  maximum  values  of  T  and  P,  A,  or  —  >  will 

—  T 

be  2.6.     If  our  construction  equations  are  made  to  read  y  = and  y  = 

100 

A  we  shall  be  within  the  customary  3"  length.     This  may  be  done  by  multiply- 
ing the  first  column  by  100  and  proceeding  as  before. 


=  o 


rnn            o         T 

-T 

(*)      (y) 
-r 

o         i    —A 

TOO         P           O 

= 

-I             I    -4 
rnr>    IP        -  P       n 

— 

TOO 

i       -i         A 
-P 

= 

IOO 

i         i           A 
P 

Then 

1    loo+P 

[    loo+P 

p 

(X 
M 

_r                             (^) 

.11 

—  I 

( 

-  A 

P) 

IOO+P 

These  equations  show  that  T  is  to  be  plotted  downward  from  the  10 
side  of  the  engineer's  scale,  that  A  is  on  a  parallel  vertical  line,  one  unit  distant 

from  T  measured  along  X ,  while  P  is  plotted  on  X  either  by  calculation  from 

p 
the  formula  x  =  —     ,  n  or  by  projection  from  the  T  scale. 

IOO+P 

Now  the  yl-axis  carries  a  scale  which  involves  the  functions  of  the  two 
variables/  and  a,  and  since  the  same  point  on  A  may  correspond  to  widely 
different  values  of /and  a  the  problem  is  to  so  graduate  the  line  that  the  effect 
of  each  variable  will  be  apparent. 

This  is  accomplished  by  constructing  an  arbitrary  network  of  lines  for 
these  two  variables,  one  set  of  lines  representing  /  and  the  other  a,  and 
arranging  them  in  such  a  manner  that  the  intersection  of  two  lines,  one  from 


THE   USE    OF   DETERMINANTS 


153 


each  set,  when  projected  on  A  will  give  the  length  on  A  corresponding  to 
these  two  values. 

For  instance  we  may  draw  a  set  of  lines  parallel  to  A  and  preferably 
at  equal  distances  apart,  numbering  them  to  correspond  to  the  different 
arcs  of  contact  we  propose  to  use  between  180°  and  270°.  Then  take  some 

IO0.0076/a 

one  value  of  /  and  solve  the  equation  A  =  — 0>0076/  a_     for  this  value  of  / 

and  a  series  of  values  of  a.  Draw  a  horizontal  from  the  foot  of  A,  and 
from  this  as  a  zero  lay  off  on  the  a  lines  the  calculated  values  of  A  in  inches. 
For  instance,  with  a  =  180°  and/  =  0.2  A  will  be  2.139".  Connect  the  points 
thus  found  by  a  smooth  curve  and  letter  it  with  the  value  of  /  used  in  the 


O-K-O 


FIG.  79. — Chart  for  brake  bands. 

calculations.  Do  the  same  with  the  other  values  of/  wanted.  Finally  draw 
a  series  of  horizontal  lines  across  the  net  to  be  used  as  guide  lines  in  pro- 
jecting. The  finished  chart  is  shown  in  Fig.  79.  To  read  it  find  the  inter- 
section of  the  chosen  f  and  a  lines,  interpolating  if  necessary,  and  project 
this  point  across  to  A .  From  this  point  on  A  draw  a  line  through  P.  Its 
intersection  with  T  will  give  the  desired  value  of  that  quantity.  The  solu- 
tion is  shown  for  the  case  when  a  =  230°, /  =  0.25,  and  P  =  50,  which  makes 
T  a  trifle  under  79. 

CONCLUDING  REMARKS. 

In  concluding  this  chapter  it  may  not  be  amiss  to  make  a  few  comments 
on  some  points  which  must  be  observed  if  we  expect  to  get  charts  which  will 
be  satisfactory  in  use. 


154  CONSTRUCTION   OF   GRAPHICAL   CHARTS 

First  in  importance  is  the  question  of  accuracy.  The  greatest  care 
should  be  exercised  in  locating  the  axes  as  well  as  in  graduating  them.  Slip- 
shod work  is  sure  to  give  unsatisfactory  results,  and  the  cause  of  such  trouble 
is  often  difficult  to  locate.  If  the  chart  does  not  give  results  which  are 
reasonably  close  to  the  calculated  values,  and  it  is  apparent  that  no  large 
error  has  been  made,  it  will  generally  save  time  to  make  a  new  chart  and 
exercise  greater  care. 

Only  a  hard,  sharp  pencil  should  be  used,  and  when  the  lines  are  inked 
great  care  should  be  taken  to  get  the  inked  line  exactly  over  the  pencil  mark 
and  not  to  one  side  of  it. 

All  straight-edges  should  be  tested  before  using  for  truth.  I  have  found 
that  a  great  deal  of  trouble  is  saved  if  they  are  made  of  steel,  including  one 
at  the  edge  of  the  drawing  board  for  the  head  of  the  T  square  to  bear  against. 

Inked  lines  should  be  made  as  fine  as  possible.  The  charts  shown  here 
are  not  intended  to  be  models  in  this  respect.  For  the  sake  of  emphasis 
the  lines  which  would  be  left  on  the  finished  chart  are  made  heavy,  and  con- 
struction lines,  which  would  be  erased,  are  made  light.  If  the  chart  is  to 
be  reproduced  by  some  photographic  process  there  is  a  Jimit  to  the  fineness  of 
the  lines,  as  they  tend  to  become  "rotten"  if  this  is  carried  too  far.  Lines 
intended  for  reproduction  should  be  a  true  black  and  not  brown,  as  is  apt  to 
be  the  case  if  the  ink  is  too  thin. 

Before  inking,  the  chart  should  be  carefully  tested  for  accuracy  in  different 
places,  as  it  is  difficult  to  make  corrections  afterward. 

The  graduating  marks  and  numbering  should  lie  on  both  sides  of  the 
axis,  otherwise  the  straight  edge  used  to  locate  the  index  line  is  likely  to  cover 
figures  or  lines  which  are  needed  for  reading.  This  difficulty  is  sometimes 
avoided  by  scratching  a  fine  line  on  a  strip  of  celluloid  or  glass,  which  is 
laid  over  the  drawing  and  used  as  an  index.  This  also  avoids  the  pencil 
marks  on  the  chart,  which,  in  time,  may  badly  deface  it. 

Where  great  accuracy  is  needed  it  is  not  safe  to  make  the  chart  on  tracing 
cloth  and  blueprint  it  As  all  draftsmen  know,  tracing  cloth  is  very  sensi- 
tive to  moisture  in  the  atmosphere,  and  the  resulting  expansion  or  shrinkage 
may  seriously  affect  the  location  of  the  scales  and  their  graduations.  If  the 
expansion  were  equal  in  all  directions  no  harm  would  be  done,  but  this  is 
not  likely  to  be  the  case.  Blueprints  are  subject  to  somewhat  similar  condi- 
tions due  to  the  necessity  for  washing  them,  but  I  have  always  felt  that  there 
was  much  less  danger  from  them  than  from  tracing  cloth. 

Charts  intended  for  photographic  reproduction  should  be  copied  with  a 
lens  which  is  truly  rectilinear,  otherwise  straight  lines  in  the  original  will  be 
reproduced  as  curves  and  accuracy  will  be  lost.  It  is  of  no  great  importance 


THE   USE    OF   DETERMINANTS  155 

that  lines  which  are  parallel  in  the  original  should  be  parallel  in  the  reproduc- 
tion, except  in  the  case  of  charts  of  the  parallel  alinement  type.  If  the  copy- 
ing board  is  not  exactly  parallel  to  the  plate  in  the  camera  the  reproduction 
will  be  a  perspective  of  the  original  but  any  points  which  were  in  alinement 
in  one  will  line  up  in  the  other. 

Finally,  every  effort  should  be  made  to  have  the  chart  "fool  proof." 
See  that  the  scales  are  arranged  in  as  simple  a  manner  as  possible.  See 
that  each  one  is  properly  designated  and  see  that  explicit  directions  are 
on  the  chart  for  its  use.  Always  give  the  formula  from  which  the  chart  is 
plotted.  Many  engineering  formulas  are  of  doubtful  value,  or  are  intended 
for  use  only  under  certain  restricted  conditions.  Sufficient  information 
should  be  given  as  to  the  basic  formulas  to  enable  the  user  to  decide  for 
himself  if  it  fits  his  case. 


INDEX 

Alinement  charts,  Chap.  II,  18,  21,  24,  26,  28,  32,  37,  41,  49,  55,  57,  61,  63,  65,  119,  122,  125, 

126,  131,  133,  137,  142,  145,  148,  151,  153 
Axonometric  charts,  100,  101,  102,  103 
Chart  for  areas,  2 

alloys,  109 

band  brakes,  8 

Bazin  formula,  148 

beams,  strength  of,  41 

bearings,  101,  102    103 

belt  area  and  I.H.P.,  70 

brake  bands,  153 

centrifugal  force,  55 

chimney  draft,  63 

circular  areas,  2,  12 

collapsing  pressure  for  tubing,  21,  86 

columns,  strength  of,  65 

connecting  rod,  72 

cylinders,  strength  of,  49,  137 

energy  of  moving  mass,  133 

falling  bodies,  law  of,  28 

fan  efficiency,  107 

flow  of  steam  through  orifices,  91 

flow  of  water  through  orifices,  126 

focal  distances  of  lens,  n,  47 

friction  drives,  77 

gas  engine,  80,  105 

gear  teeth,  strength  of,  37 

indicator  card,  gas  engine,  80 

inertia,  polar  moment  of,  61 

modulus  of  resilience,  131 

multiplication,  5,  6,  12 

piston  rod  diameter,  57 

polar  moment  of  inertia,  61 

powers  and  roots,  1 2 

resistance  of  earth  to  compression,  52 

roots  and  powers,  12 

section  modulus,  13,  119,  122 

slip  of  friction  drives,  77 

speed  of  vessel  and  I.H.P.,  74 

spherical  distance,  151 

springs,  strength  of  helical,  32,  142,  145 

twisting  moment  in  shafts,  24 

weir  formula,  125 
Determinants,  use  of,  Chap.  VIII 

157 


158  INDEX 

Double  alinement  charts,  49,  52,  55,  57,  61,  63  146,  148 
Empirical  equations,  Chap.  VI 

alinement  method,  84,  90 

curved  lines,  72 

method  of  areas  and  moments,  81 

method  of  selected  points,  75,  76 

straight  lines,  70 

Five  variable  charts,  Chap.  Ill,  37,  145 
Four  variable  charts,  Chap.  Ill,  8,  32,  41,  49,  52,  55,  57,  61,  63,  133,  137,  138,  142,  145, 151, 

iS3 

Hexagonal  index  charts,  Chap.  IV 

Logarithmic  charts,  12,  13,  18,  21,  26,  32,  37,  41,  55,  57,  80,  131,  142 
"N"  or  "Z"  type  charts,  24,  58,  61,  65,  113,  145,  153 
Parallel  alinement  charts,  49,  55,  57,  61,  63,  146,  148 
Proportional  charts,  Chap.  V 

Rectangular  coordinate  charts,  3,  4,  5,  6,  8,  n,  12,  13 
Scales,  curved,  27,  125,  126,  148 

doubled,  25 

irregular,  10 

logarithmic,  10,  128 

projected,  119 

reciprocal,  10,  123 
Single  scale  charts,  2 
Solid  models,  Chap.  VII 
Stereographic  charts,  Chap.  VII 
Triaxial  models,  109 
Two-scale  chart,  126 
"Z"  or  "N"  type  charts,  24,  58,  61,  65,  113,  145,  153 


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